pre-algebra 10-6 systems of equations 10-6 systems of equations pre-algebra homework & learning...

Pre-Algebra

10-6 Systems of Equations10-6 Systems of Equations

Pre-Algebra

HOMEWORK & Learning GoalHOMEWORK & Learning Goal

Lesson PresentationLesson Presentation

AIMS PrepAIMS Prep

PA HOMEWORK Answers

Page 521

#1-11 ALLNO WORK= ZERO CREDIT!

NO WORK= ZERO CREDIT!

Pre-Algebra

10-6 Systems of EquationsDon’t forget your proper heading! Trade & Grade!

10-5 Lesson Quiz: Part 1Solve for the indicated variable.

1. P = R – C for C.

2. P = 2l+ 2w for l.

3. V = Ah for h.

4. R = for S.

C = R - P

C – Rt = S

13C – S

t

= h3VA

= lP – 2w2

Pre-Algebra

10-6 Systems of Equations

Lesson Quiz: Part 2

5. Solve for y and graph 2x + 7y = 14.

y = – + 2 2x 7

y4

2

–2

–4

2 4–2–4

Pre-Algebra


Pre-Algebra HOMEWORK

26

#17-32 NO WORK= ZERO CREDIT!

NO WORK= ZERO CREDIT!

Pre-Algebra


Our Learning GoalStudents will be able to solve

multi-step equations with multiple variables, solve

inequalities and graph the solutions on a number line.

Pre-Algebra


Our Learning Goal Assignments• Learn to solve two-step equations.

• Learn to solve multistep equations.

• Learn to solve equations with variables on both sides of the equal sign.

• Learn to solve two-step inequalities and graph the solutions of an inequality on a number line.

• Learn to solve an equation for a variable.

• Learn to solve systems of equations.

Pre-Algebra


Today’s Learning Goal Assignment

Learn to solve systems of equations.

Pre-Algebra


Vocabulary

system of equationssolution of a system of equations

Pre-Algebra


A system of equations is a set of two or more equations that contain two or more variables. A solution of a system of equations is a set of values that are solutions of all of the equations. If the system has two variables, the solutions can be written as ordered pairs.

Pre-Algebra


Determine if the ordered pair is a solution of the system of equations below.

5x + y = 7x – 3y = 11

Additional Example 1A: Identifying Solutions of a System of Equations

A. (1, 2)

5x + y = 7

5(1) + 2 = 7?

7 = 7

x – 3y = 11

1 – 3(2) = 11 ? Substitute for

x and y.–5 11

The ordered pair (1, 2) is not a solution of the system of equations.

Pre-Algebra


Determine if each ordered pair is a solution of the system of equations below.

4x + y = 8x – 4y = 12

Try This: Example 1A

A. (1, 2)

4x + y = 8

4(1) + 2 = 8?

6 8

x – 4y = 12

1 – 4(2) = 12 ? Substitute for

x and y.–7 12


Pre-Algebra


Additional Example 1B: Identifying Solutions of a System of Equations

B. (2, –3)

5(2) + –3 = 7 ?

7 = 7

2 – 3(–3) = 11? Substitute for

x and y.11 = 11

The ordered pair (2, –3) is a solution of the system of equations.


5x + y = 7x – 3y = 11

5x + y = 7 x – 3y = 11

Pre-Algebra


Try This: Example 1B


4x + y = 8x – 4y = 12

B. (2, –3)

4(2) + –3 = 8 ?

5 8

2 – 4(–3) = 12? Substitute for

x and y.14 12

The ordered pair (2, –3) is not a solution of the system of equations.

4x + y = 8 x – 4y = 12

Pre-Algebra


Additional Example 1C: Identifying Solutions of a System of Equations

C. (20, 3)

5(20) + (3) = 7 ?

103 7

20 – 3(3) = 11? Substitute for

x and y.11 = 11



5x + y = 7x – 3y = 11

5x + y = 7 x – 3y = 11

Pre-Algebra


Try This: Example 1C

C. (1, 4)



4x + y = 8x – 4y = 12

4(1) + 4 = 8 ?

8 = 8

1 – 4(4) = 12? Substitute for

x and y.–15 12

4x + y = 8 x – 4y = 12

Pre-Algebra


When solving systems of equations, remember to find values for all of the variables.

Helpful Hint

Pre-Algebra


Additional Example 2: Solving Systems of Equations

Solve the system of equations. y = x – 4y = 2x – 9

Solve the equation to find x.

x – 4 = 2x – 9– x – x Subtract x from both sides.

–4 = x – 9

5 = x

+ 9 + 9 Add 9 to both sides.

y = x – 4 y = 2x – 9

y = y

x – 4 = 2x – 9

Pre-Algebra


Additional Example 2 Continued

To find y, substitute 5 for x in one of the original equations.

y = x – 4 = 5 – 4 = 1

The solution is (5, 1).

Check: Substitute 5 for x and 1 for y in each equation.

y = x – 4 y = 2x – 9

1 = 5 – 4? 1 = 2(5) – 9

?

1 = 1 1 = 1

Pre-Algebra


Try This: Example 2

Solve the system of equations. y = x – 5y = 2x – 8

Solve the equation to find x.

x – 5 = 2x – 8– x – x Subtract x from both sides.

–5 = x – 8

3 = x

+ 8 + 8 Add 8 to both sides.

y = x – 5 y = 2x – 8

y = y

x – 5 = 2x – 8

Pre-Algebra


Try This: Example 2 Continued

To find y, substitute 3 for x in one of the original equations.

y = x – 5 = 3 – 5 = –2

The solution is (3, –2).

Check: Substitute 3 for x and –2 for y in each equation.

y = x – 5 y = 2x – 8

–2 = 3 – 5 ? –2 = 2(3) – 8

?

–2 = –2 –2 = –2

Pre-Algebra


To solve a general system of two equations with two variables, you can solve both equations for x or both for y.

Pre-Algebra


Additional Example 3A: Solving Systems of Equations

Solve the system of equations.

A. x + 2y = 8 x – 3y = 13x + 2y = 8 x – 3y = 13

–2y –2y + 3y + 3y

Solve both equations for x.

x = 8 – 2y x = 13 + 3y

8 – 2y = 13 + 3y+ 2y + 2y

8 = 13 + 5y

Add 2y to both sides.

Pre-Algebra


Additional Example 3A Continued

8 = 13 + 5y

–13 –13

–5 = 5y

Subtract 13 from both sides.

–55

5y 5 = Divide both sides

by 5.–1 = y

x = 8 – 2y = 8 – 2(–1) Substitute –1 for y. = 8 + 2 = 10The solution is (10, –1).

Pre-Algebra


Try This: Example 3A


A. x + y = 5 3x + y = –1x + y = 5 3x + y = –1

–x –x – 3x – 3x

Solve both equations for y.

y = 5 – x y = –1 – 3x

5 – x = –1 – 3x+ x + x

5 = –1 – 2x

Add x to both sides.

Pre-Algebra


Try This: Example 3A Continued

5 = –1 – 2x

+ 1 + 1

6 = –2x

Add 1 to both sides.

Divide both sides by –2.

–3 = x

y = 5 – x = 5 – (–3) Substitute –3 for x. = 5 + 3 = 8The solution is (–3, 8).

Pre-Algebra


You can choose either variable to solve for. It is usually easiest to solve for a variable that has a coefficient of 1.

Helpful Hint

Pre-Algebra


Additional Example 3B: Solving Systems of Equations


B. 3x – 3y = -3 2x + y = -53x – 3y = –3 2x + y = –5

–3x –3x –2x –2x


–3y = –3 – 3x y = –5 – 2x

–3–3

3x–3

–3y–3 = –

y = 1 + x

1 + x = –5 – 2x

Pre-Algebra


Additional Example 3B Continued

+ 2x + 2x Add 2x to both sides.

1 + 3x = –5–1 –1

3x = –6

1 + x = –5 – 2x


–6 3

3x3 =

Divide both sides by 3.

x = –2y = 1 + x

= 1 + –2 = –1 Substitute –2 for x.

The solution is (–2, –1).

Pre-Algebra


Try This: Example 3B


B. x + y = –2 –3x + y = 2x + y = –2 –3x + y = 2

– x – x + 3x + 3x


y = –2 – x y = 2 + 3x

–2 – x = 2 + 3x

Pre-Algebra


+ x + x Add x to both sides.

–2 = 2 + 4x–2 –2

–4 = 4x

–2 – x = 2 + 3x


Divide both sides by 4.–1 = x

y = 2 + 3x= 2 + 3(–1) = –1 Substitute –1

for x.The solution is (–1, –1).

Try This: Example 3B Continued

Pre-Algebra

10-6 Systems of EquationsDon’t forget your proper heading! Trade & Grade!

10-6 Lesson Quiz

1. Determine if the ordered pair (2, 4) is a solution of the system. y = 2x; y = –4x + 12

Solve each system of equations.

2. y = 2x + 1; y = 4x

3. 6x – y = –15; 2x + 3y = 5

4. Two numbers have a some of 23 and a difference of 7. Find the two numbers.

yes

(–2,3)

15 and 8

( , 2)12

pre-algebra 10-6 systems of equations 10-6 systems of equations pre-algebra homework & learning...

Documents