grade 8 | unit 4 systems of linear equations

Copyright © 2018 Quipper Limited

1

STUDY GUIDE

GRADE 8 | UNIT 4

Systems of Linear Equations

Table of Contents

Introduction ............................................................................................................................................... 3

Test Your Prerequisite Skills ........................................................................................................ 4

Objectives ...................................................................................................................................... 5

Lesson 1: Introduction to Systems of Linear Equations

- Warm Up! ........................................................................................................................... 5

- Learn about It! ................................................................................................................... 6

- Let’s Practice! ..................................................................................................................... 8

- Check Your Understanding! ............................................................................................ 12

Lesson 2: Solution of a System of Linear Equations in Two Variables

- Warm Up! ......................................................................................................................... 13

- Learn about It! ................................................................................................................. 14

- Let’s Practice! ................................................................................................................... 17


Lesson 3: Solving Systems of Linear Equations in Two Variables: Substitution

- Warm Up! ......................................................................................................................... 23

- Learn about It! ................................................................................................................. 24

- Let’s Practice! ................................................................................................................... 25


Lesson 4: Solving Systems of Linear Equations in Two Variables: Elimination

- Warm Up! ......................................................................................................................... 32

- Learn about It! ................................................................................................................. 33


2

STUDY GUIDE

- Let’s Practice! ................................................................................................................... 34


Lesson 5: Solving Systems of Linear Equations in Two Variables: Graphing

- Warm Up! ......................................................................................................................... 42

- Learn about It! ................................................................................................................. 43

- Let’s Practice! ................................................................................................................... 46


Challenge Yourself! ..................................................................................................................... 55

Performance Task ....................................................................................................................... 56

Wrap-up ....................................................................................................................................... 59

Key to Let’s Practice! .................................................................................................................... 61

References ................................................................................................................................... 62


3

STUDY GUIDE

GRADE 8 | MATHEMATICS

UNIT 4

Systems of Linear Equations

Have you ever encountered solving an equation where two variables are not only bound

by a single constraint but by two or more? Chances are you are trying to solve a system of

linear equations.

Systems of linear equations can be illustrated in

varied real-life situations. Its usefulness and

versatility is used extensively in various sciences and

disciplines.

Often times, we encounter real-life problems where one quantity is dependent on

another quantity, but that these quantities are bound by not just a single constraint.

Uniform motion problems, mixture and work

problems, investment problems, and even

puzzle problems are examples of the

application of systems of linear equations.

This unit shall discuss how our day-to-day constraints may be

modeled mathematically through a system of linear equations. The

underlying principles regarding the solutions to these problems are

systematically discussed in this unit. Algebraic and graphical

methods to illustrate and solve these systems are discussed in

different lessons in this unit.

Click Home icon to go back to

Table of Contents


4

STUDY GUIDE

Before you get started, answer the following items to help you assess your prior

knowledge and practice some skills that you will need in studying the lessons in this unit.

1. Determine which of the following ordered pairs are solutions to the equation

.

a. (0, 15) b. (1, 11) c. (4, 3) d. (5, 4)

2. Determine the value of in for a given value of .

a. b. c.

d.

3. Transform the following equations to slope-intercept form.

a. b. c. d.

4. Complete the table of values for each given equation then graph.

a. b.

Determining whether or not an ordered pair is a solution to a linear

equation

Solving for the value of a variable in a linear equation given the value of

the other variable

Transforming linear equations from standard form to slope-intercept

form

Graphing linear equations

Modeling word problems as linear equations

Test Your Prerequisite Skills


5

STUDY GUIDE

5. Write a linear equation that would model the problem below and solve it:

Pampanga and Manila are about 240 km apart. A car leaves Manila traveling

towards Pampanga at 65 kph. At the same time, a bus leaves Pampanga bound for

Manila at 55 kph. How long will it take before they meet?

At the end of this unit, you should be able to recognize systems of linear equations;

identify if a point is a solution to a system of linear equations or not;

enumerate the different types of solutions of a system of linear equations in two

variables;

define each category which a system of linear equation could fall into;

identify, without graphing, which system of linear equations has parallel,

intersecting, or overlapping lines; and

solve a system of linear equations in two variables by substitution, graphing, and

elimination.

Picture It Clearly!

Materials Needed: activity sheet, ruler, coloring and drawing materials

Instructions:

1. The activity may be done by pair or by group.

Lesson 1: Introduction to Systems of Linear Equations

Objectives

Warm Up!


6

STUDY GUIDE

2. Your teacher will provide an activity sheet in the form of an image half the

size of a legal bond paper.

3. The image or figure may be in the form of a work of art, a landscape,

infrastructure, etc.

4. Construct a rectangular coordinate plane on the image and label it

accordingly.

5. From the image, find elements that illustrate each mathematical concept

below.

lines points

on a line

parallel

lines

slope of a

line points

intersecting

lines abscissa

graph of

a line

coordinates

of points

linear

equations

6. Describe each concept using the format given below.

“My idea of (state the mathematical concept) is

________________________________”

Some of the terms you encountered and illustrated in the Warm Up! activity, words like

graph of a line, linear equations, slope of a line, and points, will recurrently be used in this

lesson and the succeeding ones. Reviewing prior knowledge on these will be helpful in

understanding the concepts in this unit.

Let us now discuss what a system of linear equations is.

Learn about It!


7

STUDY GUIDE

A pair of lines whether parallel, intersecting, or coinciding illustrates a system of linear

equations.

The degree of the variables in linear equations is either 0 or 1 and no two variables are

being multiplied with each other. So, if you see or in the equation, then it is not

linear.

The graph of a system of linear equations will involve two or more lines.

Consider the following system of linear equations in two variables and its graph:

Definition 1.1: A system of linear equations involves two or

more linear equations.

It may have the form

, , , , and are constants, while and ,

and and , must not be both equal to zero.


8

STUDY GUIDE

A solution to a system of linear equations in two variables is an ordered pair for

which the values of and satisfy each of the equations in the system.

Graphically, these two values are represented as the point of intersection, with ordered

pair , of both lines.

Example 1: Give an example of a system of linear equations in one variable.

Solution: Answers may vary but since the instruction says the equation needs to be in

one variable, we can use either or . Let us say, for example, we use .

One possible answer would be

Note that the variable in both equations is of degree 1 or 0 (specifically,

both are of degree 1).

Try It Yourself!

Give an example of a system of linear equations in two variables.

Definition 1.2: The point of intersection of two lines is

the point where the two lines meet.

Let’s Practice!


9

STUDY GUIDE

Example 2: Show that the point ( is a solution of the given system of linear equations

in two variables.

Solution:

Step 1: Identify the values of and from the given point.

Given the point , and .

Step 2: Substitute the values of and into each of the equations in the system, and

then simplify.

Equation 1:

The point satisfies the first equation.

Equation 2:

The point satisfies the second equation.

Therefore, the point is the solution of the system

since it

satisfies both equations.


10

STUDY GUIDE

Try It Yourself!

Is the point a solution of the given system of linear equations in two

variables?

Example 3: Determine whether is a solution of the system of equations

and .

Solution:

Step 1: Identify the values of and from the given point.

Given the point , and .

Step 2: Substitute the values of and into each of the equations in the system, and

then simplify.

Equation 1:

The point satisfies the first equation.


11

STUDY GUIDE

Equation 2:

The point satisfies the second equation.

Therefore, the point (3, 2) is the solution of the system

since it

satisfies both equations.

Try It Yourself!

Determine whether or not is a solution of the system of equations

and .

Real-World Problems

Example 4: An exam contains 20 questions. Some items are worth 2

points and the rest are worth 3 points. The exam is worth

100 points. Construct a system of linear equations that

models the situation.

Solution:

Step 1: Define the variables.

Let be the number of two-point questions.

Let be the number of three-point questions.


12

STUDY GUIDE

Step 2: Translate English phrases or statements into mathematical expressions or

equations.

An exam contains 20 questions. ⇒

Some items are worth 2 points ⇒

The rest are worth 3 points ⇒

The exam is worth 100 points. ⇒

Therefore, the system of linear equations that models the situation is

.

Try It Yourself!

A jeepney driver charges ₱8 for passengers who rode for

less than 4 km and ₱10 for passengers who rode for

more than 4 km. The jeepney has a total capacity of 20

passengers, which makes him earn ₱180. Construct a

system of linear equations that models the situation

where is the number of passengers who rode for less than 4 km and is the

number of passengers who rode for more than 4 km.

1. Give two examples of a system of linear equations in two variables.

2. Determine if the point is a solution to the system of linear equations

Check Your Understanding!


13

STUDY GUIDE

3. The admission fee at a theme park is ₱300 for children and ₱500 for adults. A

family of ten entered the theme park and paid ₱2,600. Construct a system of linear

equations that represent the problem.

What Do They Look Like?

Materials Needed: graphing paper, ruler, pencil or any drawing materials

Instructions:

1. The activity may be done by groups of three or five.

2. Your group will be given a system of linear equations, and you are to graph each

equation by changing it to slope-intercept form.

3. Complete the table below to guide you in graphing each equation, then answer the

questions that follow.

Equation Slope y-Intercept

1.

2.

a. What is the slope of equation 1? of Equation 2?

b. What is the y-intercept of equation 1? of Equation 2?

c. What can you observe with the slope and y-intercept of equations 1 and 2?

Lesson 2: Solution of a System of Linear Equations in

Two Variables

Warm Up!


14

STUDY GUIDE

d. What relationship can you infer regarding the slope, y-intercept, and the graph

of the system?

Below are example of pairs of equations that may be used for the activity.

a. and –

b. and

c. and

One way to determine the graph of a system of linear equations is to compare the linear

equations with each other in terms of their slopes and y-intercepts. To do this, as you may

have observed in Warm Up!, it is best to express both linear equations in the slope-

intercept form , where is the slope and is the y-intercept, and check

whether the slopes and y-intercept follow the conditions below;

If the slopes of the two lines are equal, then the graph of the system consists of

either as a set of parallel lines (not equal y-intercepts) or overlapping lines (equal y-

intercepts).

If the slopes are not equal, then the graph of the system consists of a set of

intersecting lines.

Systems of linear equations are also categorized according to the type of solutions.

Learn about It!


15

STUDY GUIDE

A consistent system can further be identified as either dependent or independent.

All solutions to one of the equations in a dependent system are also solutions to the other

equation.

Recall that the graph of a linear equation in the form is a line, hence, two

such equations can be graphed with two lines on the same rectangular or Cartesian

coordinate plane.

The three types of solutions also have varying graphs.

A consistent-dependent system has a graph that consists of overlapping lines.

Definition 2.4: An independent system has only one

solution.

Definition 2.3: A dependent system has infinitely many

solutions.

Definition 2.2: A system of linear equations is said to be

inconsistent if it has no solution.

Definition 2.1: A system of linear equations is said to be

consistent if it has either one unique or more

than one (infinitely many) solution(s).


16

STUDY GUIDE

A consistent-independent system has a graph that consists of intersecting lines.

The graph of an inconsistent system consists of parallel lines.


17

STUDY GUIDE

Example 1: Given the system

, what must be the value of for the system

to be inconsistent?

Solution: An inconsistent system is a system that has no solution because the graph of

the system are parallel lines. Since the graphs are parallel lines, therefore the

slopes must be equal and the y-intercepts must be unequal. Since the given

system already has unequal y-intercepts, must be unequal to .

Try It Yourself!

Given the system

, what must be the value of and for the system to

be consistent and independent?

Example 2: Identify if the system of linear equations

will produce

intersecting, parallel or overlapping lines.

Step 1: Express each equation in slope-intercept form .

Equation 1:

Let’s Practice!


18

STUDY GUIDE

Equation 2:

Step 2: Find the slope and the y-intercept of each equation.

Equation 1:

;

Equation 2:

;

Step 3: Compare the slopes and the y-intercepts.

Since the slopes are equal, the graph of the system will form either parallel

or overlapping lines. But since the y-intercepts are equal too, the graph

consists of overlapping lines.

Thus, you may conclude that the system of linear equations

has infinitely many solutions, and its graph consists of

overlapping lines.

Try It Yourself!

Identify if the system of linear equations

will produce intersecting,

parallel or overlapping lines.


19

STUDY GUIDE

Example 3: Identify if the given system of linear equations

will produce

intersecting, parallel, or overlapping lines.

Solution:

Step 1: Express each equation in slope-intercept form.

Equation 1:

Equation 2:

Step 2: Find the slope and the y-intercept of each equation.

Equation 1:

;

Equation 2:

;

Step 3: Compare the slopes and the y-intercepts.

Since the slopes are not equal, this system will form intersecting lines.


20

STUDY GUIDE

Therefore, we may conclude that the system of linear equations

has a graph consisting of intersecting lines, and has only

one solution.

Try It Yourself!

Identify if the given system of linear equations

will produce

intersecting, parallel, or overlapping lines.

Real-World Problems

Example 4: Joanna is asked to look for two unknown numbers. The

sum of the two numbers is 87. Twice the smaller number

less the larger number is 18. Which among the two pairs

below can be the two unknown numbers?

a. 43 and 44 b. 35 and 52

Solution:

Step 1: Represent the smaller number as and the larger number as .

Let be the smaller number.

be the larger number.

Step 2: Represent the system.

The sum of the two numbers is 87

Twice the larger number less the smaller number is 18.


21

STUDY GUIDE

Thus, the system would be

Step 3: Since both pairs of numbers add up to 87, we need to verify if difference of

twice the smaller less the larger is 18.

Given 43 and 44.

Given 35 and 52.

Therefore, 35 and 52 are solutions of the system of equations.

Try It Yourself!

Can 32 and 55 be a solution for the two unknown numbers in

Example 4? Justify your answer.


22

STUDY GUIDE

1. Identify if the given system of linear equations will produce intersecting, parallel, or

overlapping lines.

a.

b.

c.

2. Identify the type of system for each of the given systems in Number 1 as either

consistent-independent, consistent-dependent, or inconsistent.

3. Janice is thinking of two integers. The smaller integer is and the greater integer is

. When she gets the value of , she obtains . What do you think will

be equal to?



23

STUDY GUIDE

If I Were, Then U Are

Materials Needed: fishbowl, pen, paper

Instructions:

1. This activity may be done by pair or by group.

2. Your teacher has prepared a fishbowl with rolled small pieces of papers.

3. A member of each group will then pick a piece of paper inside the fishbowl and

then will move at the back of the classroom.

4. You will then be given time to read the question in each piece of paper.

5. Your teacher will then give a signal for you to solve and answer the question on the

board.

6. When you write your solution on the board, you have to write the given and write

your answer in the form “If I were ____, then U are _____.”

7. Your teacher will then verify your answers.

8. The group of the player who gives the right answer first gets a point.

9. The group with the most points wins.

Example of questions:

Given: If I were 4, then you are ______.

Given: If I were 3, then you are ______.

Given: If I were -1, then you are ______.

Lesson 3: Solving Systems of Linear Equations in Two

Variables: Substitution

Warm Up!


24

STUDY GUIDE

One method to solve a system of linear equations is the substitution method.

In Warm Up!, the substitution method was used to find the value of , however, to solve a

system of linear equation using the substitution method, one of the equations is to be

manipulated to be expressed in one variable in terms of the other variable. This derived

value would then be substituted to the second equation, forming a new equation with

only one variable.

Consider the following system of linear equations:

What values of and satisfy the system?

To solve for and by substitution, the following steps may be helpful:

Step 1: Manipulate one equation so that one variable is in terms of the other.

Note: For convenience, choose an equation with a single variable or one with a

variable whose coefficient is 1.

Solve for in terms of .

Step 2: Substitute the derived expression into the second equation.

Learn about It!


25

STUDY GUIDE

Step 3: Solve the equation.

Step 4: Substitute the derived value in Step 3 into the equation in Step 1.

Thus, the final answers are and . This solution can also be written as the

point .

Example 1: Solve the following system of linear equations using substitution.

Let’s Practice!


26

STUDY GUIDE

Solution:

Step 1: Since Equation 2 is already expressed in terms of , we can quickly substitute

it into Equation 1.

Step 2: Substitute the expression in Step 1 into the first equation.



Thus, the solutions are and . This solution can also be written as

the ordered pair .

Try It Yourself!

Solve the following system of linear equations using substitution.


27

STUDY GUIDE

Example 2: Solve the following system of linear equations using substitution.

Solution:


Note: For convenience, choose an equation with a single variable or one with a

variable whose coefficient is 1.

The first equation has a with a coefficient of 1.



Step 3: Solve the resulting equation.


28

STUDY GUIDE



the ordered pair .

Try It Yourself!


Example 3: Solve the system of linear equations.

Solution:




29

STUDY GUIDE





the ordered pair .

Try It Yourself!



30

STUDY GUIDE

Real-World Problems

Example 4: Go Green! purchased 51 ornamental trees to plant in a

park. They bought a number of small and large trees. If

the number of the small trees is twice the number of the

large trees, how many trees of each size did they plant?

Solution:

Step 1: Represent the number of small trees as and the number of large trees as .

Let be the large trees

be the small trees


There are 51 trees in all.

The number of small trees is twice the number of large trees.

Thus, the system would be

Step 3: Since equation 2 is already expressed in terms of , we can substitute the

value of to the first equation.


31

STUDY GUIDE

Step 4: Substitute the derived value in Step 3, to

Therefore, there are 17 large trees and 34 small trees.

Try It Yourself!

The length of the top of a rectangular table is 5 more than twice its

width. Its perimeter is 70 cm. What is its area?

1. Solve the following systems of linear equations by substitution.

a.

b.

2. Analyze and solve the following problem:

There were 48 students who auditioned for a school’s Glee Club. There were three

times as many female students who auditioned as there were males. How many

males and how many females auditioned for the school’s Glee Club?



32

STUDY GUIDE

Neutral Pairs

Materials Needed: paper and pen, black and white cartolinas

Instructions:

1. This activity may be done individually or in pairs.

2. Create algebra tiles using the black and white cartolinas.

3. Label the tiles accordingly: x-tiles are the small rectangular tiles, y-tiles are the big

rectangular tiles while the small square tiles are 1 unit each.

4. Given and , model the equations using the algebra tiles.

5. Combine the two equation models.

6. Remove the neutral pairs and write the new equation.

7. Answer the following questions:

a. Which variable was removed?

b. What is the resulting equation when the variable was removed?

c. Can you solve the system using these method? Justify your answer.

8. Present your findings in class.


Variables: Elimination

Warm Up!


33

STUDY GUIDE

Another way to solve systems of linear equations algebraically, besides substitution, is

through elimination of variables. In elimination, either or is removed or eliminated

leading to an equation in one variable. The variable is retained when is eliminated or

removed and vice-versa. Using the algebra tiles in the Warm Up! activity is one way of

showing the elimination of variables.

Algebraically, elimination is a technique for solving systems of linear equations which

involves cancelling, or eliminating, one variable by adding or subtracting the two

equations.

A key step in the process is multiplying either or both of the equations by a constant such

that the coefficients for either or have the same absolute value but opposite signs.

To understand it better, consider the following system of linear equations:

Which variable has coefficients with the same absolute value?

The coefficients of the variable have the same absolute value; that is, 1 and –1. Hence, to

eliminate , the two equations must be added.

Learn about It!


34

STUDY GUIDE

Solving for , we have

The derived value for can now be substituted into any of the given equations to find .

Therefore, the solution to the system is .

Example 1: Solve the following system using elimination:

Solution:

Step 1: Choose a variable to eliminate.

By inspection, it is easier to eliminate instead of since the coefficients of

in both equations have the same absolute value and

Step 2: Add the two equations to eliminate the chosen variable.

Let’s Practice!


35

STUDY GUIDE

Step 3: Solve the resulting linear equation.

Step 4: Find the value of the second variable.

Substitute the value of the variable you found to any of the original

equations.

Thus, the solution of the system is and . This solution may also

be expressed as the ordered pair .

Try It Yourself!

Solve the following system of linear equations using elimination:


36

STUDY GUIDE


Solution:


Let us say, we choose to eliminate .

Step 2: Manipulate the equations so that the coefficients of the chosen variable in

both equations have the same absolute value but different signs.

Multiply the first equation by because from the second equation has a

coefficient of 2.



Step 5: Find the value of the second variable.


37

STUDY GUIDE


equations.

Thus, the solution of the system is and . This may be

expressed as the ordered pair

Try It Yourself!



Solution:


It can be seen that it is easy to find a multiplier for in the second equation

to have the same absolute value as the coefficient of in the first equation.

Thus, we choose to eliminate .


38

STUDY GUIDE

Step 2: Manipulate the equations so that the coefficients of the chosen variable in

both equations have the same absolute value but of opposite signs.

Multiply the second equation by 10.



Step 5: Find the value of the other variable.


equations.

–

Thus, the solution of the system is and . This may be expressed

as the ordered pair .


39

STUDY GUIDE

Try It Yourself!


Real-World Problems

Example 4: Soybean meal is 16% protein while cornmeal is 9% protein. How

many pounds of each should be mixed together to get a 350 lb

mixture that is 12% protein?

Solution:

Step 1: Let number of pounds of soybean meal.

number of pounds of cornmeal.

Step 2: List important information in a table.

Soybean Cornmeal Mixture

Amount of meal

to be mixed

Percent of

protein

Amount of

protein in

mixture

Note: 42 was obtained by getting 12% of 350.


40

STUDY GUIDE


Note: The system may be transformed into the following by multiplying

Equation 2 by 100 for easier computation:

Step 4: Start solving the system by multiplying the first equation by to eliminate

the variable .


Step 4: Solve for .


41

STUDY GUIDE

Thus, the mixture should contain 150 pounds of soybean meal and 200

pounds of cornmeal.

Try It Yourself!

Solution A is 2% alcohol and solution B is 6% alcohol. A drugstore

owner wants to mix the two solutions to get a 60-liter solution that is

3.2% alcohol. How many liters of each solution should the owner use?

1. Solve the following systems of linear equations by elimination:

a.

b.

2. Analyze and solve the following problem:

A jet can travel 1200 kilometers in 4 hours with the wind. The return trip against the

wind takes 6 hours. Find the rate of the jet in still air and the rate of the wind.



42

STUDY GUIDE

It’s Graphing Time!

Materials Needed: smartphone or tablet with a graphing application, pen, paper,

graphing paper, ruler (optional)

Instructions:

1. This activity may be done in groups of 3.

2. Using the graphing application in your smartphone or tablet, input the linear

equations listed. You may seek the assistance of your teacher in doing so if you are

not familiar with the application. If the use of electronic gadgets is not allowed, you

may opt to graph the equations in a large piece of graphing paper.

3. After graphing the equations, observe the relationship between the graphs of the

given equations then complete the table that follows.

Equations:

Equation 1:

Equation 2:

Equation 3:

Equation 4:

Equation 5:

Equation 6:


Variables: Graphing

Warm Up!


43

STUDY GUIDE

Equations What can you

say about

their graphs?

Is there a

common

point?

What type of

system is

given?

How many

solutions are

there?

Equations 1

and 2

Equations 3

and 4

Equations 5

and 6

Aside from algebraic methods like substitution and elimination, another way of finding

the solution of a system of linear equations in two variables is by graphing or by the

graphical method. Solving a system of linear equations in two variables by graphing will

result in three cases as observed in the Warm Up! activity.

1. If the graphs of the two equations form intersecting lines (consistent and

independent), then the system has a unique solution.

Learn about It!


44

STUDY GUIDE

2. If the two equations represent the same line; that is, the graphs form overlapping

lines (dependent), then the system has infinitely many solutions.

3. If the graphs of the two equations form parallel lines (inconsistent), then the

system has no solution.

In graphing an equation, it is best to rewrite it first into the slope-intercept form

, where represents the slope, and is the y-intercept.

The following systems of linear equations are graphed to determine the solution.

1. Unique solution


45

STUDY GUIDE

The point of intersection is (−1, 3). This point represents the solution to the system.

2. Infinitely many solutions

This is a case of overlapping lines. Thus, the system has infinitely many solutions.

This means all the points on the line are solutions to the system.

3. No solution


46

STUDY GUIDE

The graph shows two parallel lines. Note that they do not intersect. Thus, this

system has no solution.

Example 1: Find the solution set of the system of linear equations shown below by

graphing using their x and y intercepts.

Solution:

Step 1: Determine the x and y-intercepts of each equation in the given system.

For

If , then The y-intercept is .

Let’s Practice!


47

STUDY GUIDE

If , then The x-intercept is .

For

If , then The y-intercept is .

If , then The x-intercept is .

Step 2: Graph each line using the intercepts.

Plot the intercepts and draw a straight line through both points.

Step 3: Identify the solution based on the graph.

The graph is a pair parallel lines. Thus, the system has is inconsistent. It has

no solution.


48

STUDY GUIDE

Try It Yourself!

Find the solution set of the system of linear equations shown below by graphing.


graphing.

Solution: This time, we shall use the slope-intercept form in graphing.

Step 1: Graph the first linear equation.

Graph using the slope and the y-intercept.

The slope is −2, and the y-intercept is 4, or (0, 4).

Thus, from the point (0, 4), go down 2 units, then count 1 unit to the right to

get the second point (1, 2). Connect the two points to form a line.


49

STUDY GUIDE

Step 2: Graph the second linear equation.

Graph using intercepts.

Find the x-intercept:

The x-intercept is .

Find the y-intercept:

The y-intercept is .


50

STUDY GUIDE

Plot the intercepts and draw a straight line through both points.


The graph is a set of parallel lines. Thus, the system has no solution.

Try It Yourself!

Find the solution set of the system of linear equations shown below by graphing

using their x- and y-intercepts.


graphing.

Solution:


51

STUDY GUIDE

Step 1: Graph the first linear equation.

Graph using the slope and the y-intercept. Rewrite the equation in slope-

intercept form.

The slope is −2, and the y-intercept is −4, or

Step 2: Graph the second linear equation.


52

STUDY GUIDE

Graph using the slope and the y-intercept. Rewrite the equation in slope-

intercept form.

The slope is −3, and the y-intercept is 1, or .


The graph is a set of lines intersecting at Thus, the system has a

unique solution: and .

Try It Yourself!

Find the solution set of the system of linear equations shown below by graphing.


53

STUDY GUIDE

Real-World Problems

Example 4: A total of ₱25 000 is invested in a bank in two funds

paying 6% and 8% annual interest. The combined

annual interest is ₱1800. How much of the ₱25 000 is

invested in each fund?

Solution:

Step 1: Let x be the amount invested in 6% annual interest.

y be the amount invested in 8% annual interest.

Step 2: List important information in a table.

Total Amount

Annual Interest

Total Interest Earned

Step 3: Represent the system

Note: The system may be transformed into the following by multiplying

Equation 2 by 100 for easier computation:


54

STUDY GUIDE

Step 4: Graph the system.

Equation 1 has intercepts at (25000, 0) and (0, 25000).

Equation 2 has intercepts at (30000, 0) and (0, 22500).

The two lines intersect at (10000, 15000), thus, the amount invested at 6% is

₱10 000 while the amount invested at 8% interest rate is ₱15 000.

Try It Yourself!

A total of ₱50 000 is invested in two funds paying 5% and 6% annual interest. The

annual interest is ₱2 800. Find how much of the ₱50 000 pesos is invested in each?


55

STUDY GUIDE

1. Find the solution set of the system of linear equations shown below by graphing.

a.

b.

2. On a municipal hospital, about 51 babies are born every day. Of these babies, there

are about twice as many girls as there are boys. How many boys and girls are born

each day?

1. If a system of linear equations may be solved through graphing, why do you think

there is a need to know the algebraic methods (substitution and elimination)? On

what instances do you think is it more convenient to use the substitution method

over elimination method, and vice versa?

2. Write an equation which can be paired to the equation to form a

system with a solution of .

Challenge Yourself!



56

STUDY GUIDE

3. Find the system of linear equations with the following graph.

This activity will showcase your learning in this unit. You will assume the role of a travel

specialist/advisor working in a travel company.

The company is launching their new travel packages for various types of customers. As a

specialist, you are tasked to provide detailed travel packages and plans in the form of

brochures highlighting the different travel destinations in Luzon. Specifically, you are to

provide camping and hiking destinations as well as computations for camping materials

and consumables. The materials include tents, ropes, cooking utensils, as well as other

items you may think are necessary. Food packages are also provided for the

campers/travelers in an optional basis.

You may refer to the given format below as guide for your brochure.

Performance Task


57

STUDY GUIDE

I. Travel Destination

Location

Short Description

Images

II. Basic Travel Packages for:

Single (₱2000 per pax)

Couples (₱1500 per pax)

Family (₱1000 per pax)

Friends/Barkada (₱1000 per pax)

Team Building/Company (₱750 per pax)

III. Materials and Food

You are going to present the brochure and plan to the head of your travel agency for

approval. Once approved, the brochure will then be printed for circulation and marketing

purposes. As part of the planning you must provide data for the following inquiries below

prior to the presentation.

1. List of all camping materials needed. Specify the quantity for each as well as the

price. Keep in mind that the materials should be appropriate based on the travel

package and the needs of the travelers (e.g. adults, kids, male or female)

2. List of all ingredients needed for the menu for the food packages. Specify the

quantities needed and the unit price for each ingredient. Keep in mind that the

ingredients should be appropriate based on the travel package and the needs of

the travelers (e.g. adults, kids, male or female)

3. The data can be presented in tabular form. You may refer to the following table as

your guide.


58

STUDY GUIDE

TOUR PACKAGE MATERIALS FOOD INGREDIENTS

NAME QUANTITY COST NAME QUANTITY COST

4. Use the data from the table to formulate at least 5 problems involving systems of

linear inequalities in two variables then solve each problem.

The brochure and plan will be evaluated according to the following: design and

presentation, accuracy, usefulness, and mathematical justification.

Performance Task Rubric

Criteria

Below

Expectation

(0–49%)

Needs

Improvement

(50–74%)

Successful

Performance (75–

99%)

Exemplary

Performance

(99+%)

Design and

Presentation

The design and

presentation is

poor.

The design and

presentation is

somewhat

informative.

The design and

presentation is

informative and

flawless.

The design and

presentation is

very informative

and flawlessly

done. It is also

easy to

understand.

Accuracy

The information

given are

erroneous and do

not show wise use

The information

given are

erroneous and

show some use of

The information

given are accurate

and shows use of

the concepts of

The information

given are accurate

and show a wise

use of the


59

STUDY GUIDE

of the concepts of

systems of linear

equations in two

variables.

the concepts of

systems of linear

equations in two

variables.

systems of linear

equations in two

variables.

concepts of

systems of linear

equations in two

variables.

Usefulness

The brochure is

not useful in

understanding

the information

given.

The brochure is

somewhat useful

in understanding

the information

given.

The brochure is

well-crafted and

useful for

understanding the

information given.

It showcases the

proper layout.

The brochure is

well-crafted and

useful for

understanding the

information given.

It showcases the

proper layout and

is accurately done.

Mathematical

Justification

Justification is

ambiguous. Only

few concepts of

systems of linear

equations in two

variables are

applied.

Justification are

not so clear. Some

ideas are not

connected to each

other. Not all

concepts of

systems of linear

equations in two

variables are

applied.

Justification is

clear and

informatively

delivered.

Appropriate

concepts learned

on systems of

linear equations in

two variables are

applied.

Justification is

logically clear,

informative, and

professionally

delivered. The

concepts learned

on systems of

linear equations in

two variables are

applied.

Systems of Linear Equations in Two

Variables

Solving Systems of Linear Equations in

Two Variables

Algebraic Method:

Elimination and

Substitution

Graphical Method

Wrap-up


60

STUDY GUIDE

Key Terms/Formulas

Concept Relevant Information

System of Linear

Equations

A system of linear equations involves two or more linear

equations.

The solution of a system of linear equations in two

variables involves a value of x and y that satisfies both

equations.

Graphically, the solution is the point of intersection of the

two lines.

Solutions of System of

Linear Equations

A consistent system of two linear equations has one or

more solutions. A consistent system can be either

dependent or independent.

A dependent system has infinitely many solutions. All

solutions for one of the equations are also solutions to the

other equation. Its graph consists of two overlapping lines.

An independent system has one solution. Its graph

consists of two intersecting lines.

An inconsistent system has no solution. Its graph consists

of two parallel lines.

Solving Systems of

Linear Equations by

Substitution

The substitution method is a technique in solving system

of linear equations that expresses one variable in terms of

the other variable. This derived expression would then

be substituted to the second equation, forming a new

equation with only one variable.

Solving Systems of

Linear Equations by

Elimination

Elimination is a technique for solving systems of linear

equations that involves cancelling, or eliminating, one

variable by adding or subtracting the two equations.


61

STUDY GUIDE

Solving Systems of

Linear Equations by

Graphing

Graphing is one method to solve a system of linear

equations in two variables.

A system with a unique solution is represented by the

point of intersection of two lines.

A system with no solution is represented by two parallel

lines.

A system with infinitely many solutions is represented by

overlapping lines.

Lesson 1

1.

, or answers may vary.

2. Yes

3. No

4.

Lesson 2

1. and can be any number as long as

2. intersecting lines

3. overlapping lines

4. No, because the pair of numbers given cannot satisfy the two equations.

Lesson 3

1.

2.

3.

4. 250 sq. units

Key to Let’s Practice!


62

STUDY GUIDE

Lesson 4

1.

2.

3.

4. 42 liters of solution A and 18 liters of solution B

Lesson 5

1. The graph is a set of parallel lines. Thus, the system has no solution.

2. The graph is a set of lines intersecting at . Thus, the system has a unique

solution: and .

3. The system has infinitely many solutions since the graph of the lines are

overlapping.

4. The amount invested at 5% interest is ₱20,000 while the amount invested at 6%

interest is ₱30,000.

Abuzo, Emmanuel P., et al. Mathematics Learners’ Module Grade 8. Book Media Press Inc.,

2013

Baron, Lorraine, et al. Math Makes Sense 8. Canada: Pearson Education, 2008.

Maths Is Fun. “Algebra”. Accessed February 28, 2018.

http://www.mathsisfun.com/algebra/systems-linear-equations.html

McGraw-Hill Education. Glencoe Math Volume 1. McGraw-Hill Professional, 2013.

Oronce, Orlando A., et. al. E-Math Intermediate Algebra. Philippines: Rex Bookstore, Inc.

2007

References

grade 8 | unit 4 systems of linear equations

Documents