Please help! Algebra 2 3x3 equation

Question

A football team scored a total of 50 points in one game.

They scored 14 times.

The scoring was made up of touchdowns (6 points each), points after touchdown (1 point each) and field goals (3 points each).

They had three more touchdowns than field goals. How many of each type of score did the team have?

Please help! I am stuck, and don't know what to do, thanks!

Answer 1

This problem is a system of three equations with three unknown variables. We have to first come up with the correct equations, then we will solve them.

Let x = # of touchdowns scored (worth 6 pts each)

Let y = # of points after touchdown scored (worth 1 pt each)

Let z = # of field goals scored (worth 3 pts each)

The first sentence of the problems states the team scored 50 points in total. We can write an equation for this as follows:

6x + y + 3z = 50 (call this equation #1)

The next sentence states they scored 14 times. We can write an equation for this as follows:

x + y + z = 14 (equation #2)

Finally, we know the team had three more touchdowns than field goals. This means if we had 1 field goal, there would be 4 touchdowns. 2 field goals would be 5 touchdowns. A general way to write this equation in terms of our variables is as follows:

x = z + 3 (equation #3)

There are many ways to solve these equations. To me, the simplest way is as follows. We will substitute the term for x in equation #3 directly into equations #1 and #2.

6*(z + 3) + y + 3z = 50

6z + 18 + y + 3z = 50

9z + y + 18 = 50

9z + y = 32

Now we will do the same for equation #2.

(z + 3) + y + z = 14

2z + y + 3 = 14

2z + y = 11

We can multiply our now simplified equation #2 by a factor of -1, then we can add it to equation #1.

-2z - y = -11

9z + y = 32 (add these two lines together, to get the resulting line below)

7z = 21

z = 3

Now, we can substitute our answer for z back into equation #3.

x = z + 3

x = 3 + 3

x = 6

Finally, we can substitute our answers for x and z back into the original equation #2.

x + y + z = 14

6 + y + 3 = 14

y = 5

To summarize, the team scored 6 touchdowns, 5 points after touchdown, and 3 field goals. You can verify this answer is correct by substituting into our original equations. All three will come out true.