Question

question
stella is a high school basketball player. in a particular game, she made some free throws (worth one point each) and some two point shots. stella scored a total of 12 points and made twice as many free throws as two point shots. graphically solve a system of equations in order to determine the number of free throws made, x, and the number of two point shots made, y.

Explanation:

Step1: Define Variables and Equations

Let \( x \) be free throws (1 pt each), \( y \) be two - point shots (2 pts each). From total points: \( x + 2y=12 \). From the relationship of number of shots: \( x = 2y \).

Step2: Rewrite Equations in Slope - Intercept Form

For \( x + 2y=12 \), solve for \( y \):
\( 2y=-x + 12 \), so \( y=-\frac{1}{2}x + 6 \).
For \( x = 2y \), solve for \( y \): \( y=\frac{1}{2}x \).

Step3: Graph the Equations

• The line \( y = \frac{1}{2}x \) has a slope of \( \frac{1}{2} \) and y - intercept at \( (0,0) \).
• The line \( y=-\frac{1}{2}x + 6 \) has a slope of \( -\frac{1}{2} \) and y - intercept at \( (0,6) \).

Step4: Find the Intersection

To find the intersection, we can also solve the system algebraically (since we know the intersection is the solution). Set \( \frac{1}{2}x=-\frac{1}{2}x + 6 \).
Add \( \frac{1}{2}x \) to both sides: \( \frac{1}{2}x+\frac{1}{2}x=6 \), \( x = 6 \).
Substitute \( x = 6 \) into \( x = 2y \), we get \( 6 = 2y \), so \( y = 3 \).
(Graphically, the two lines intersect at \( (6,3) \), where \( x \) is the number of free throws and \( y \) is the number of two - point shots.)

Answer:

The number of free throws \( x = 6 \) and the number of two - point shots \( y = 3 \).