Using Algebra to Solve Linear Systems8-8

Warm UpWarm Up

Lesson PresentationLesson Presentation

Problem of the DayProblem of the Day

Lesson QuizzesLesson Quizzes

Using Algebra to Solve Linear Systems8-8

Warm UpSolve for the indicated variable.

1. P = R – C for R

2. V = Ah for A

3. R = for C

R = P + C

Rt + S = C

13C – S

t

= A3Vh

Using Algebra to Solve Linear Systems8-8

Problem of the Day

At an audio store, stereos have 2 speakers and home-theater systems have 5 speakers. There are 30 sound systems with a total of 99 speakers. How many systems are stereo systems and how many are home-theater systems?

17 stereo systems, 13 home-theater systems

Using Algebra to Solve Linear Systems8-8

MA.8.A.1.3 Use tables, graphs, and models to represent, analyze, and solve real-world problems related to systems of linear equations.

Sunshine State Standards

Using Algebra to Solve Linear Systems8-8

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

Caution!

Using Algebra to Solve Linear Systems8-8

Additional Example 1A: Solving Systems of Equations

Solve the system of equations.

y = 4x – 6

y = x + 3

y = 4x – 6 y = x + 3

The expressions x + 3 and 4x – 6 both equal y. So by the Transitive Property they are equal to each other.

4x – 6 = x + 3

Using Algebra to Solve Linear Systems8-8

Additional Example 1A Continued

To find y, substitute 3 for x in one of the original equations.y = x + 3 = 3 + 3 = 6

The solution is (3, 6).

Solve the equation to find x.

4x – 6 = x + 3 – x – x Subtract x from both sides.

3x – 6 = 3

3x 9 6 6 Add 6 to both sides.

3 = 3 x = 3

Divide both sides by 3.

Using Algebra to Solve Linear Systems8-8

The system of equations has no solution.

2x + 9 = –8 + 2x – 2x – 2x

Transitive Property

Subtract 2x from both sides. 9 ≠ –8

Additional Example 1B: Solving Systems of Equations

y = 2x + 9y = –8 + 2x

Using Algebra to Solve Linear Systems8-8

Check It Out: Example 1A

Solve each system of equations.

y = x – 5y = 2x – 8

x – 5 = 2x – 8 y = x – 5

3 = x y = (3) – 5 = –2

The solution is (3, –2).

Using Algebra to Solve Linear Systems8-8

The solution is (6, 12).

2x = x + 6

Check It Out: Example 1B

y = 2xy = x + 6

x = 6

y = 2x

y = 2(6) = 12

Solve each system of equations.

Using Algebra to Solve Linear Systems8-8

When equations in a system are not already solve for one variable, you can solve both equations for x or both for y.

Using Algebra to Solve Linear Systems8-8

Additional Example 2A: Solving Systems of Equations by Solving for a Variable

Solve the system of equations.

x + 4y = –10 x – 3y = 11x + 4y = –10 x – 3y = 11

–4y –4y 3y 3y

Solve both equations for x.

x = –10 – 4y x = 11 + 3y

– 3y – 3y–10 – 7y = 11

Subtract 3y from both sides.

–10 – 4y = 11 + 3y

–10 – 4y = 11 + 3y

Using Algebra to Solve Linear Systems8-8

Additional Example 2A Continued

+10 +10

– 7y 21

Add 10 to both sides.

Divide both sides by –7. –7 = – 7

x = 11 + 3y = 11 + 3(–3) Substitute –3 for y. = 11 + –9 = 2The solution is (2, –3).

–10 – 7y = 11

y = –3

Using Algebra to Solve Linear Systems8-8

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

Helpful Hint

Using Algebra to Solve Linear Systems8-8

Additional Example 2B: Solving Systems of Equations by Solving for a Variable

Solve the system of equations.

–2x + 10y = –8 x – 5y = 4–2x + 10y = –8 x – 5y = 4 –10y –10y +5y +5y

Solve both equations for x.

–2x = –8 – 10y x = 4 + 5y

= ––8–2

10y–2

–2x–2

x = 4 + 5y4 + 5y = 4 + 5y

– 5y – 5ySubtract 5y from both sides.4 = 4

Since 4 = 4 is always true, the system of equations has an infinite number of solutions.

Using Algebra to Solve Linear Systems8-8

Check It Out: Example 2A

Solve each system of equations.

2x + y = 0

2x + 3y = 8

y = −2x and y =

–2x = ; x –2

2x + y = 0

−2x + 83

−2x + 83

2(−2) + y = 0

y = 4The solution is (–2, 4).

Using Algebra to Solve Linear Systems8-8

Check It Out: Example 2B

Solve the system of equations.

y = x –1

–3x + 3y = 4

y = x − 1 and y =

x – 1 =

−3 ≠ 4

3x + 43

3x + 43

There are no solutions.

Using Algebra to Solve Linear Systems8-8

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

Using Algebra to Solve Linear Systems8-8

Lesson Quiz

Solve each system of equations.1. y = 5x + 10

y = –7 + 5x

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

3. 6x – y = –15 2x + 3y = 5

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

(–2, 3)

15 and 8

( , 2)12

no solution

Using Algebra to Solve Linear Systems8-8

1. Solve the given system of equations.

y = 11x + 20

y = –2 + 11x

A. (2, 2)

B. (1, 1)

C. (1, –1)

D. no solution

Lesson Quiz for Student Response Systems

Using Algebra to Solve Linear Systems8-8

2. Solve the given system of equations.

4x + y = 11

2x + 3y = –7

A. (4, –5)

B. (4, 5)

C. (2, –5)

D. (2, 5)

Lesson Quiz for Student Response Systems

Using Algebra to Solve Linear Systems8-8

3. Two numbers have a sum of 37 and a difference of 17. Identify the two numbers.

A. –27 and –10

B. –27 and 10

C. 27 and 10

D. 27 and –10

Lesson Quiz for Student Response Systems