Q30. - Algebra 2

Question

In the 2000-2001 season, Minnesota’s Katie Smith was ranked first in the WNBA for total points and three-point field goals made. She scored 646 points making 355 shots, including 3-point field goals, 2-point field goals, and 1-point free throws. She made 27 more 2-point field goals than 3- point field goals.

Find the number of each type of goal she made.

Step-by-Step Solution

Answer

She made 88 three-point field goals, 115 two-point field goals, and 152 one point free throws.

Step 1 – First construct a system of equations corresponding to the given situation.

Let $x$ be the number of 1-point free throws, $y$ be the number of 2-point field goals, and $z$ be the number of 3-point field goals.

 In the figure, it is given that the total 355 shots are made.

So, $x+y+z=355$.

 In the figure, it is given that the total scored points is 646.

So, $x+2y+3z=646$

 In the figure, it is given that the 27 more 2-point field goals than 3-point field goals.

This can be mathematically written as $y=z+27\Rightarrow y-z=27$.

So, the system of equations will be:

$\begin{array}{d}x+y+z=355\\ x+2y+3z=646\\ y-z=27\end{array}$

Step 2 – Use the elimination method to get the system of equations in two variables.

Add $x+y+z=355$  to the equation $y-z=27$.

$\begin{array}{l}\underset{\_}{\begin{array}{l}x+y+z=355\\ y-z=27\end{array}}\\ x+2y+0=382\end{array}$

So, the resultant equation is $x+2y=382$

Multiply the$equation y-z=27$  by 3 and add the new resultant equation to the equation $x+2y+3z=646$

$\begin{array}{l}\underset{\_}{\begin{array}{l}y-z=27\\ x+2y+3z=646\end{array}}\begin{array}{c}\text{multiply by 3}\\ \end{array}\underset{\_}{\begin{array}{l}3y-3z=81\\ x+2y+3z=646\end{array}}\\ x+5y+\text{\hspace{0.17em}\hspace{0.17em}}0=727\end{array}$

.So, the resultant equation is $x+5y=727$.

.

Step 3 – Solve the system of equations in two variables

Subtract $x+2y=382$ from$x+5y=727$ as follows:

$\begin{array}{l}\underset{\_}{\begin{array}{l}x+5y=727\\ (-)x+2y=382\end{array}}\\ 3y=345\end{array}$

Solve $3y=345$ for $y$:

$\begin{array}{c}3y=345\\ \frac{3y}{3}=\frac{345}{3}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} divide both sides by }3\\ y=115\end{array}$

Step 4 – Find the values of x and z .

Substitute $y=115 in x+2y=382$ and find the value of $x$

$\begin{array}{c}x+2y=382\\ x+2\left(115\right)=382\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Substitute }115\text{ for }y\\ x+230=382\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Simplify}\\ x=152\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Subtract }230\text{ from both sides}\end{array}$

Substitute $y=115 in y-z=27$ and find the value of $z$

$\begin{array}{c}y-z=27\\ 115-z=27\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Substitute }115\text{ for }y\\ -z=-88\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Subtract }115\text{ from both sides}\\ z=88\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Divide both sides by -1}\end{array}$

.Hence, the solution of the given system of equations is$\left(x,y,z\right)=\left(152,115,88\right)$.

So, she made 88 three-point field goals, 115 two-point field goals, and 152 one point free throws.