# Thread: Setting up a system of equations! + Matrix solving

1. ## Setting up a system of equations! + Matrix solving

In a recent football game, KU scored 32 points in 9 scoring plays (from touchdowns worth 6 points each, extra points worth 1 point each, and field goals worth 3 points each), and got one more touchdown than extra point. Set up a system of equations to determine how many touchdowns, extra points, and field goals they made, then put this system into a matrix and s solve using row operations.

2. Hello, ryno16!

In a recent football game, KU scored 32 points in 9 scoring plays
(Touchdowns = 6 points, extra points = 1 point, field goals = 3 points.)
And got one more touchdown than extra points.

Set up a system of equations to determine how many TDs, EPs, and FGs they made,
then put this system into a matrix and solve using row operations.

Let: .$\displaystyle \begin{Bmatrix}T &=& \text{touchdowns} \\ E &=& \text{extra points} \\ F &=& \text{field goals} \end{Bmatrix}$

"32 points in 9 plays": .$\displaystyle \begin{array}{ccc}T + E + F &=& 9 \\ 6T + E + 3F &=& 32 \end{array}$

"One more T than E": .$\displaystyle T \:=\:E+1 \quad\Rightarrow\quad T - E \:=\:1$

We have: .$\displaystyle \left|\begin{array}{ccc|c} 1&1&1&9 \\ 6&1&3&32 \\ 1&\text{-}1&0 & 1 \end{array}\right|$

$\displaystyle \begin{array}{c} \\ R_2-6R_1 \\ R_3-R_1\end{array}\left|\begin{array}{ccc|c} 1&1&1&9 \\ 0&\text{-}5& \text{-}3& \text{-}22 \\ 0& \text{-}2& \text{-}1 & \text{-}8 \end{array}\right|$

. . $\displaystyle \begin{array}{c}\\ \text{-}1\cdot R_2 \\ \text{-}1\cdot R_3 \end{array} \left|\begin{array}{ccc|c} 1&1&1&9 \\ 0&5&3&22\\ 0&2&1&8 \end{array}\right|$

$\displaystyle \begin{array}{c}\\ R_2 - 2R_3 \\ \\ \end{array} \left|\begin{array}{ccc|c} 1&1&1&9 \\ 0&1&1&6 \\ 0&2&1&8 \end{array}\right|$

$\displaystyle \begin{array}{c}R_1-R_2 \\ \\ R_3-2R_2 \end{array} \left|\begin{array}{ccc|c} 1&0&0&3 \\ 0&1&1&6 \\ 0&0&\text{-}1 & \text{-}4 \end{array}\right|$

. . $\displaystyle \begin{array}{c} \\ \\ \text{-}1\cdot R_3\end{array} \left|\begin{array}{ccc|c} 1&0&0&3 \\ 0&1&1&6 \\ 0&0&1&4 \end{array}\right|$

$\displaystyle \begin{array}{c}\\ R_2-R_3 \\ \\ \end{array} \left|\begin{array}{ccc|c} 1&0&0&3 \\ 0&1&0&2 \\ 0&0&1&4 \end{array}\right|$

Therefore: .$\displaystyle \begin{Bmatrix}\text{3 touchdowns} \\ \text{2 extra points} \\ \text{4 field goals} \end{Bmatrix}$

3. thank you so much! this helped an extreme amount!