Question

engaging families

1. edward and william play a game in which they earn the same number of points for each goal and lose the same number of points for each penalty. edward makes 6 goals and 3 penalties, ending the game with 6 points. william earns 8 goals and 9 penalties and ends the game with -22 points.

a. write a system of equations that describes edward and william’s outcomes. use x to represent the number of points for a goal and y to represent the number of points for a penalty.

b. solve the system by using substitution. what does your solution mean?

Explanation:

Step1: Define equations for Edward

Edward's total points: $6x - 3y = 6$

Step2: Define equations for William

William's total points: $8x - 9y = -22$

Step3: Simplify Edward's equation

Divide by 3: $2x - y = 2$, rearrange to $y = 2x - 2$

Step4: Substitute $y$ into William's equation

Substitute $y=2x-2$:
$8x - 9(2x - 2) = -22$
Expand: $8x - 18x + 18 = -22$
Simplify: $-10x = -40$
Solve for $x$: $x = 4$

Step5: Solve for $y$

Substitute $x=4$ into $y=2x-2$:
$y = 2(4) - 2 = 6$

Answer:

a. The system of equations is:
$6x - 3y = 6$
$8x - 9y = -22$

b. $x=4$, $y=6$. This means each goal is worth 4 points, and each penalty subtracts 6 points.