Clare and Noah 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. Clare makes 6 goals and 3 penalties, ending the game with 6 points. Noah earns 8 goals and 9 penalties and ends the game with -22 points.
Write a system of equations that describes Clare and Noah's outcomes. Use x to represent the number of points for a goal and to y to represent the number of points for a penalty.
Solve the system. What does your solution mean?
what would y equal?
Let's break down the information given in the problem:
For Clare:
- Clare makes 6 goals and ends the game with 6 points.
- She also has 3 penalties.
For Noah:
- Noah makes 8 goals and ends the game with -22 points.
- He has 9 penalties.
We can now write the system of equations:
For Clare:
6x - 3y = 6 ----(1)
For Noah:
8x - 9y = -22 ----(2)
To solve this system of equations, we can use any method, such as substitution or elimination. I will use the method of elimination.
Multiplying equation (2) by 2, we get:
16x - 18y = -44 ----(3)
Now, if we add equations (1) and (3), we can eliminate the x terms:
(6x - 3y) + (16x - 18y) = 6 + (-44)
Simplifying:
22x - 21y = -38 ----(4)
Now, we have two equations with two variables:
(4) -> 22x - 21y = -38
(2) -> 8x - 9y = -22
We can solve this new system using elimination or substitution to find the values of x and y.
Multiplying equation (4) by 8 and equation (2) by 22, we get:
(4) -> 176x - 168y = -304 ----(5)
(2) -> 176x - 198y = -484 ----(6)
Now, we subtract equation (5) from equation (6) to eliminate the x terms:
(6) - (5) -> (176x - 198y) - (176x - 168y) = -484 - (-304)
Simplifying:
-30y = -180
Solving for y:
y = -180 / -30
y = 6
Now, we substitute the value of y back into equation (2) to solve for x:
8x - 9(6) = -22
Simplifying:
8x - 54 = -22
8x = 54 - 22
8x = 32
x = 32 / 8
x = 4
Therefore, the solution to the system of equations is x = 4 and y = 6.
In this context, x represents the number of points earned for a goal, and y represents the number of points lost for a penalty. Thus, Clare earns 4 points for each goal, and loses 6 points for each penalty, while Noah earns 4 points for each goal and loses 6 points for each penalty.
This means that Clare gains 4 points for each goal she makes and loses 6 points for each penalty she commits. Noah also gains 4 points for each goal he makes, but loses 6 points for each penalty he commits.
To create a system of equations, we can use the given information and assign variables to represent the number of points for a goal and a penalty. Let's use x to represent the number of points for a goal and y to represent the number of points for a penalty.
For Clare, we know that she made 6 goals and 3 penalties, ending the game with 6 points. This can be represented by the equation:
6x - 3y = 6
For Noah, we know that he made 8 goals and 9 penalties, ending the game with -22 points. This can be represented by the equation:
8x - 9y = -22
Now we have a system of equations:
6x - 3y = 6
8x - 9y = -22
To solve this system, we can use any method such as substitution or elimination. Let's use the substitution method.
From equation 1, we can isolate x:
6x = 6 + 3y
x = 1 + (1/2)y
Substitute this value of x into equation 2:
8(1 + (1/2)y) - 9y = -22
8 + 4y - 9y = -22
-5y = -30
y = 6
Now substitute the value of y back into equation 1 to solve for x:
6x - 3(6) = 6
6x - 18 = 6
6x = 24
x = 4
The solution to the system of equations is x = 4 and y = 6. This means that each goal is worth 4 points and each penalty is worth 6 points.
The solution means that Clare earns 4 points for each goal scored and loses 6 points for each penalty. Noah also earns 4 points for each goal scored but loses 6 points for each penalty. However, he ends up with a negative score of -22 because he made more penalties than goals.
Point per goal --- x
points per penalty --- y
Clare: 6x + 3y = 6 ---> 2x + y = 2 OR y = 2 - 2x
Noah: 8x + 9y = -22
using substituion:
8x + 9(2-2x) = -22
8x + 18 - 18x = -22
-10x = -40
x = 4
then y = 2 - 2(4) = -6
your turn to show what that means.