Question

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 -42 points. Write equations and solve for how much goals and penalties are worth.

Answer 1

goal=6

panalty= -10

Explanation:

6g+3p=6. <---- multipilly by -3

8g+9p= -42

-18g - 9p = -18

8g+9p=-42

= -10g = -60

goals are worth 6 points

than you would plug the 6 back in either equation

6(6)+3p= 6

36 + 3p = 6

-36. -36

3p=-30

p=-10

so panalties are worth -10 points

Answer 2

a) The equations for the situation are:

Clare: 6g - 3p = 6

Noah: 8g - 9p = -42.

b) The solutions to the equations show that each goal is worth 26 points, and each penalty is worth 50 points.

Clare Noah

Goals earned 6 8

Penalties earned 3 9

Ending points 6 -42

Let the number of points earned for each goal = g

Let the number of points lost for each penalty =p

Equations:

Clare: 6g - 3p = 6

Noah: 8g - 9p = -42

Solving these equations simultaneously to find the values of g and p:

First, let's solve the first equation for g:
`[ 6g = 6 + 3p ] [ g = 1 + (1)/(2)p ]`

Now, substitute this expression for g into the second equation:
`[ 8(1 + (1)/(2)p) - 9p`

= -42 ] [ 8 + 4p - 9p

= -42 ] [ -p = -50 ]

p = 50

Now that we have the value of ( p ), we can substitute it back into the expression for g:
`[ g = 1 + (1)/(2)(50) ]`

g = 26

Thus, each goal is worth 26 points, and each penalty is worth 50 points.