Systems of Equations7-4

Warm Up

Lesson PresentationProblem of the Day

Lesson Quizzes

Systems of Equations7-4

Warm UpSolve for the indicated variable.

1. P = R – C for R2. V = Ah for A3. R = for C

R = P + C

Rt + S = C

13C – S

t

= A3Vh

Systems of Equations7-4

Problem of the DayAt 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

Systems of Equations7-4

Learn to solve systems of equations.

Systems of Equations7-4

Vocabularysystem of equationssolution of a system of equations

Systems of Equations7-4

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.

Systems of Equations7-4

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

Caution!

Systems of Equations7-4

Additional Example 1A: Solving Systems of EquationsSolve the system of equations.

y = 4x – 6y = 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

Systems of Equations7-4Additional Example 1A Continued

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

Systems of Equations7-4

The system of equations has no solution.

2x + 9 = –8 + 2x – 2x – 2x

Transitive PropertySubtract 2x from both sides. 9 ≠ –8

Additional Example 1B: Solving Systems of Equations

y = 2x + 9y = –8 + 2x

Systems of Equations7-4Check It Out: Example 1A

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

y = x – 5 y = 2x – 8

x – 5 = 2x – 8

The expressions x – 5 and 2x – 8 both equal y. So by the Transitive Property they equal each other.

Systems of Equations7-4Check It Out: Example 1A Continued

To find y, substitute 3 for x in one of the original equations.y = x – 5 = 3 – 5 = –2The solution is (3, –2).

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.

Systems of Equations7-4

The system of equations has no solution.

3x – 7 = 6 + 3x – 3x – 3x

Transitive PropertySubtract 3x from both sides.

–7 ≠ 6

Check It Out: Example 1B

y = 3x – 7y = 6 + 3x

Systems of Equations7-4

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

Systems of Equations7-4Additional Example 2A: Solving Systems of Equations by

Solving for a Variable

Solve the system of equations. 5x + y = 7 x – 3y = 11

5x + y = 7 x – 3y = 11 – y – y + 3y + 3y

Solve both equations for x. 5x = 7 – y x = 11 + 3y

– 15y – 15y55 = 7 – 16y

Subtract 15y from both sides.

5(11 + 3y)= 7 – y 55 + 15y = 7 – y

Systems of Equations7-4Additional Example 2A Continued

–7 –7 48 – 16y

Subtract 7 from both sides.Divide both sides by –16.–16 = – 16

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

55 = 7 – 16y

–3 = y

Systems of Equations7-4

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

Helpful Hint

Systems of Equations7-4Additional Example 2B: Solving Systems of Equations by

Solving for a VariableSolve 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

–210y–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.

Systems of Equations7-4Check It Out: Example 2A

Solve the system of equations.

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

–x –x – 3x – 3xSolve both equations for y.y = 5 – x y = –1 – 3x

5 – x = –1 – 3x+ x + x

5 = –1 – 2x

Add x to both sides.

Systems of Equations7-4Check It Out: Example 2A Continued

5 = –1 – 2x+ 1 + 16 = –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).

Systems of Equations7-4Check It Out: Example 2B

Solve the system of equations. x + y = –2 –3x + y = 2

x + y = –2 –3x + y = 2– x – x + 3x + 3x

Solve both equations for y.

y = –2 – x y = 2 + 3x

–2 – x = 2 + 3x

Systems of Equations7-4

+ x + x Add x to both sides.

–2 = 2 + 4x–2 –2–4 = 4x

–2 – x = 2 + 3x

Subtract 2 from both sides.

Divide both sides by 4.–1 = x

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

for x.The solution is (–1, –1).

Check It Out: Example 2B Continued

Systems of Equations7-4

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

Systems of Equations7-4Lesson 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

Systems of Equations7-4

1. Solve the given system of equations. y = 11x + 20y = –2 + 11x A. (2, 2)B. (1, 1) C. (1, –1)D. no solution

Lesson Quiz for Student Response Systems

Systems of Equations7-4

2. Solve the given system of equations. 4x + y = 112x + 3y = –7 A. (4, –5)B. (4, 5)C. (2, –5)D. (2, 5)

Lesson Quiz for Student Response Systems

Systems of Equations7-4

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

A. –27 and –10B. –27 and 10C. 27 and 10D. 27 and –10

Lesson Quiz for Student Response Systems