SOLUTION: a basketball team recently scored a total of 80 points on a combination of 2-point field goals, 3-point field goals, and 1-point foul shots. Altogether, the team made 45 baskets an

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Question 876364: a basketball team recently scored a total of 80 points on a combination of 2-point field goals, 3-point field goals, and 1-point foul shots. Altogether, the team made 45 baskets and 19 more 2-pointers than foul shots. How many shots of each kind were made?
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
you have two equation to work with.

let:

x = number of 2 point shots.
y = number of 3 point shots.
z = number of 1 point shots.

since the total number of baskets is equal to 45, this leads to the equation of:

x + y + z = 45

since the total points is equal to 80, this leads to the equation of:

2x + 3y + z = 80

you are given that the number of 2 point shots is 19 more than the number of 1 point shots.

since x equals the number of 2 point shots and z equals the number of 1 point shots, this leads to the equation of:

x = z + 19

solve this equation for z to get:

z = x - 19

you can now replace z with x - 19 in both of those equations.

the attached picture below shows the details of the equations and the processing that follows.

in that picture, the following occurs:

number 1 shows you the original equations.

number 2 shows you that x = z + 19 which becomes z = x - 19 when you solve for z.

number 3 shows you the original equations after you replaced z with x - 19.

number 4 shows you the result of combining like terms and adding 19 to both sides of each of those equations. the equations in number 4 are the equations that need to be solved simultaneously.

number 5 shows you the result of multiplying both sides of the first equation by 3 and then subtracting the second equation from the modified first equation. the result of this operation is that the y variables cancel out and you are left with an equation of 3x = 93. solving for x gets you x = 31.

number 6 solves for z now that x is known by using the equation of z = x - 19.
that gets you z = 12

number 7 replaces x and z with their respective values in the first equation and then solves for y which is equal to 2.

you now have the values for all 3 variables.
you have:
x = 31
y = 2
z = 12

number 8 uses those values in the original equations to confirm the solution is good. since 45 = 45 and 80 = 80 are both true equations, the confirmation has been made and your solution is confirmed as good.