Main Idea and New Vocabulary

NGSSS

Example 1: One Solution

Example 2: Real-World Example

Example 3: Real-World Example

Example 4: No Solution

Example 5: Infinitely Many Solutions

Five-Minute Check

• Solve systems of equations by graphing.

• system of equations

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

MA.8.A.1.4 Identify the solution to a system of linear equations using graphs.

One Solution

Solve the system y = 3x – 2 and y = x + 1 by graphing.

Graph each equation on the same coordinate plane.

Check y = 3x – 2 y = x + 1

2.5 = 3(1.5) – 2 2.5 = (1.5) + 1

2.5 = 2.5 2.5 = 2.5

The graphs appear to intersect at (1.5, 2.5). Check this estimate by replacing x with 1.5 and y with 2.5.

One Solution

Answer: The solution of the system is (1.5, 2.5).

A. (2, 0)

B. (0, 3)

C. (–2, 2)

D. (–2, 4)

Solve the system y = – x + 3 and y = –x + 2 by

graphing.

PENS Ms. Baker bought 14 packages of red and green pens for a total of 72 pens. The red pens come in packages of 6 and the green pens come in packages of 4. Write a system of equations that represents the situation.

One Solution

Let x represent packages of red pens and y represent packages of green pens.

x + y = 14 The number of packages of pens is 14.

6x + 4y = 72 The number of pens equals 72.

Answer: The system of equations is x + y = 14 and 6x + 4y = 72.

A. 2x + 3y = 23x + y = 5

B. x + y = 23x + y = 9

C. 2x + 3y = 23x + y = 9

D. 2x + 3y = 9x + y = 23

FOOD Abigail and her friends bought some tacos and burritos and spent $23. The tacos cost $2 each and the burritos cost $3 each. They bought a total of 9 items. Write a system of equations that represents the situation.

One Solution

PENS Ms. Baker bought 14 packages of red and green pens for a total of 72 pens. The red pens come in packages of 6 and the green pens come in packages of 4. The system of equations that represents the situation is x + y = 14 and 6x + 4y = 72. Solve the system of equations. Interpret the solution.

Write each equation in slope-intercept form.

x + y = 14 6x + 4y = 72

y = –x + 14 4y = –6x + 72

y = – x + 18

Both equations have the same value when x = 8 and y = 6.

You can also graph both equations on the same coordinate plane.

Choose values for x that could satisfy the equations.

One Solution

One Solution

Answer: The equations intersect at (8, 6). The solution is (8, 6). This means that Ms. Baker bought 8 packages of red pensand 6 packages of green pens.

A. (4, 5); They bought 4 tacos and 5 burritos.

B. (4, 5); They bought 4 burritos and 5 tacos.

C. (5, 4); They bought 5 tacos and 4 burritos.

D. (6, 3); They bought 6 tacos and 3 burritos.

FOOD Abigail and her friends bought some tacos and burritos and spent $23. The tacos cost $2 each and the burritos cost $3 each. They bought a total of 9 items. The system of equations that represents the situation is x + y = 9 and 2x + 3y = 23. Solve the system of equations. Interpret the solution.

Solve the system y = 2x – 1 and y = 2x by graphing. Graph each equation on the same coordinate plane.

No Solution

The graphs appear to be parallel lines. Since there is no coordinate point that is a solution of both equations, there is no solution for this system of equations.

No Solution

Answer: no solution

A. (2.5, –1.5)

B. (–2.5, –6.5)

C. no solution

D. infinitely many solutions

Solve the system y = –x – 4 and x + y = 1 by graphing.

Infinitely Many Solutions

Solve the system y = 3x − 2 and y − 2x = x − 2 by graphing.

Write y − 2x = x − 2 in slope-intercept form.

y – 2x= x – 2 Write the equation.

y – 2x + 2x= x – 2 + 2x Add 2x to each side.

y = 3x – 2 Simplify.

Both equations are the same. Graph the equation.

Infinitely Many Solutions

Answer: infinitely many solutions

Any ordered pair on the graph will satisfy both equations. So, there are infinitely many solutions of the system.

A. (0, 2)

B. (2, –1)

C. no solution

D. infinitely many solutions

Solve the system y = – x + 2 and 3x + 2y = 2 by

graphing.

A. j + k = 468 C. j + k = 468 j = 52 + k j = 52 – k

B. j + k = 468 D. j – k = 468 k = 52 + j j = 52 + k

Last year, Justin and his sister, Karin, earned a total of $468 in allowance. If Justin earned $52 more than Karin in allowance, write a system of equations that represents their allowances.

A. 10x + 3x = y C. 10 + 3x = y 6x + 5x = y 6 + 5x = y

B. 10x + 3 = y D. –10 + 3x = y 6x + 5 = y –6 + 5x = y

Mrs. Kung spent the same amount on two programs at the local recreation center. The aerobics class costs an initial fee of $10 plus $3 per class. The pottery class costs an initial fee of $6 plus $5 per class. Write a system of equations to represent the cost for the two programs.

A. d + 3a = 323d − a = 4

B. 3d + a = 32 a − d = 4

C. d + 3a = 32d − a = 4

D. d + 3 + a = 32d − a = 4

The sum of Dewan’s age and three times Adrianne’s age is 32. The difference between Dewan’s age and Adrianne’s age is 4. Which system can be used to find Dewan’s age and Adrianne’s age?

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