Solving Equations

An Opportunity for Practice

Introduction

Equations are one of the most important tools used in Algebra. Equations are mathematical statements where 2 expressions are separated by an equal sign (=).

ex. 4x - 7 = 8x - 5

Most equations contain at least one unknown value denoted by a variable. Variables are symbols, usually letters, used to represent an unknown value.

ex. In the above equation, the variable is x.

Types of EquationsWe will study several different equations throughout our Algebra I and Algebra II classes.

1) Equations in 1 variable.

ex. 4x - 7 = 8x - 5

4) Linear equations in 2 variables

ex. y = 2x - 1

5) Quadratic Equations

ex. x2 + 7x = -12

2) Proportions

ex. 5

3

1

x

x

3) Fractional Equations

ex. 3

1

58

xx

Do not try to solve these equations now! We’ll explore each topic individually.

Equations in 1 VariableEx. 4x - 7 = 8x - 5

Think about these equations like a balanced scale. Your job is to get the variable by itself without disrupting the balance of the scale. The way to do this is use INVERSE (OPPOSITE) OPERATIONS.

4x - 7 = 8x - 51) We need to get the variable on 1 side. We have +8x on the right side. The opposite of positive is negative. Therefore, we’ll subtract 8x from each side.-8x -8x

-4x - 7 = -5 2) To eliminate -7, we need to add 7 to each side.

+7 +7

-4x = 23) Remember: -4x means “-4 times x”. The opposite of multiplication is division. We are going to divide both sides by –4.4- 4-

x = -½ Don’t forget to reduce your final answer!

Practice– Equations in 1 Variable

Solve each of the following equations using inverse operations. Be sure the equations stay balanced– if you do an operation on one side of the equal sign, you must do it on the other!

63)1(23)4

521)3

5223)2

273)1

xxx

xx

xx

x

**Be careful with # 4– simplify the left side first!

Once you’ve solved each equation, click the

mouse again to check the solutions!

4)4

6)3

2)2

3)1

x

x

x

x

ProportionsA proportion is an equation that sets two ratios (fractions) equal to each other.

The key to solving proportions is to rewrite it as an equation in 1 variable. We do this using a process called cross multiplying.

5

3

1

x

x 1) To cross multiply, multiply the top (numerator) of one fraction by the bottom (denominator) of the other.

2) Take the solutions and set them equal to each other.335

)1(35

xx

xx

3) Using inverse operations, solve the equation.

2

11

2

3

2 2

32

3 3

orx

x

xx

Practice-- Proportions

Solve each of the following proportions by cross multiplying.

3

7

12

4)3

6

4

30

100)2

5

4

20)1

x

x

xAfter solving each of the proportions, click your

mouse again to see the

solutions. 32)3

5)2

16)1

x

x

x

Fractional EquationsFractional equations are exactly what the name implies– equations that contain fractions.

Fractions can be a difficult concept, especially when working with equations. Our goal is to eliminate the fractions from the equation.

253

xx 1) To eliminate the fractions we have to find a common

denominator– the number that can be divided evenly by all the denominators in the equation. (In this case, the denominators are 3 and 5!)

** The smallest number that can be divided by 3 and 5 is 15. A quick way to find a common denominator is to multiply your denominators!

2) Multiply every term on both sides of the equation by the common denominator. Simplify each term.

3035

2155

153

15

xx

xx

3) Solve the Equation in 1 Variable.4

33or

4

15

8 8

308

x

x

Practice– Fractional Equations

Solve each of the following equations by finding the common denominator.

2

1

5

2

10

3)3

3

2

64)2

2

5

2

15

3

5)1

x

xx

x Once you have solved the equations, click your

mouse again to see the

solutions.2

1)3

8)2

3)1

x

x

x

Linear Equations in 2 VariablesEquations that contain 2 different variables are called linear because their graph is a line.

Unlike equations with only 1 variable, linear equations have an infinite number of solutions represented by ordered pairs (x,y). It is impossible to list all of them. Therefore, we represent the solution to linear equations by graphing.

12 xy

The easiest way to graph linear equations is to be sure the equation is in “y form”. Then, identify the slope and y-intercept*.

*For more information on slope, y intercept, and graphing, click on this link: Slope of a Line.ppt

Slope = 2 y-intercept = -1

Each point on the line can be represented by an ordered pair with an x and y value. Every point on that line satisfies (makes the statement true) our equation.

Practice– Linear EquationsIdentify the slope and y-intercept for each of the following equations. Then draw the graph to represent all of the solutions. Be sure the equation is in y-form first!

53)1 xy

xy2

1)2

2)3 yx

After you graph each of the

equations, click your mouse again to see the graph. Note the color of the line matches the color of the

equation!

1) Slope = 3 y-int = 5

2) Slope = ½ y-int = 0

3) Slope = -1 y-int = 2

Quadratic Equations—Method 1

Quadratic Equations are distinguished from other equations because they contain an “x2” term.

There are 2 methods to solve quadratic equations. Both methods require the same initial step: THE EQUATION MUST EQUAL 0! We call this “standard form”.

x2 + 7x = -12

To write the above equation in standard form, add 12 to each side.

x2 + 7x + 12 = 0

(x + 3)(x + 4) = 0

x + 3 = 0 x + 4 = 0

2) Set factors = 0.

ZERO PRODUCT PROPERTY

1) Factor the polynomial**.

x = -3 x = -4 3) Solve the equations.

**For factoring practice click this link: Factoring Expressions.ppt

Quadratic Equations– Method 2

Not all quadratic equations can be solved using zero product property because not all expressions can be factored.

QUADRATIC FORMULA

a

acbbx

cbxax

2

4

0

2

2

x2 + 7x + 12 = 0a = 1 b = 7 c = 12

)1(2

)12)(1(477 2 x

2

48497

2

17

2

17

x = -3 or -4

The key to quadratic formula is identifying a, b, and c. Then, plug the values into the formula!

Practice– Quadratic Equations

Solve each of the following quadratic equations using your method of choice.

074)3

524)2

124)1

2

2

2

xx

xx

xx

32.1,32.5)3

3,8)2

2,6)1

xx

xx

xxAfter solving,

click your mouse again to

check your solutions.

Summary

1 Variable Unique variable Use inverse operations “keep the scale

balanced”

Proportions 2 fractions equal Cross multiply

Fractional Contain at least 1

fraction Multiply all terms by

common denominator

Linear equations Contain x and y Infinite number of

solutions represented as ordered pairs

Must be in y-form Graph solution (line)

Quadratic equations Contain x2

2 methods to solve: zero product property or quadratic formula

Must be in standard form (= 0)

Mixed PracticeWe have seen many different types of equations. See if you can determine which type of equation is listed below, then solve the equation!

32)56

1

4

3)4

1025)3

02712)2

53

2

6)1

2

yx

xx

xx

yy

xxClick your

mouse again to check your solutions!

3int

2 )57

2)4

2)3

9,3)2

1)1

- y-

-slope

x

x

xx

x