Question

analyzing relationships you lose track of how many 2 - point baskets and 3 - point baskets a team makes in a basketball game. the team misses all the 1 - point baskets and still scores 54 points. the equation 2x + 3y = 54 models the total points scored, where x is the number of 2 - point baskets made and y is the number of 3 - point baskets made.
a. find and interpret the intercepts.
b. can the number of 3 - point baskets made be odd? explain your reasoning.
c. graph the equation. find two more possible solutions in the context of the problem.
multiple representations in exercises 29 - 32, match

Explanation:

Step1: Find x - intercept

Set \(y = 0\) in \(2x+3y = 54\). Then \(2x=54\), so \(x = 27\). Interpretation: If the team makes no 3 - point baskets (\(y = 0\)), they made 27 two - point baskets to score 54 points.

Step2: Find y - intercept

Set \(x = 0\) in \(2x + 3y=54\). Then \(3y=54\), so \(y = 18\). Interpretation: If the team makes no 2 - point baskets (\(x = 0\)), they made 18 three - point baskets to score 54 points.

Step3: Analyze if y can be odd

The equation is \(2x+3y = 54\), which can be rewritten as \(2x=54 - 3y\). The left - hand side \(2x\) is always even. The right - hand side \(54-3y\) must be even. Since 54 is even, \(3y\) must be even for \(54 - 3y\) to be even. Since 3 is odd, \(y\) must be even. So the number of 3 - point baskets made cannot be odd.

Step4: Rewrite the equation for graphing

Rewrite \(2x+3y = 54\) as \(y=-\frac{2}{3}x + 18\). We can find possible solutions by choosing non - negative integer values of \(x\) and \(y\) that satisfy the equation. Let \(x = 3\), then \(2\times3+3y=54\), \(6 + 3y=54\), \(3y=48\), \(y = 16\). Let \(x=6\), then \(2\times6+3y=54\), \(12+3y=54\), \(3y = 42\), \(y = 14\).

Answer:

a. x - intercept: \((27,0)\), means 27 two - point baskets and 0 three - point baskets. y - intercept: \((0,18)\), means 0 two - point baskets and 18 three - point baskets.
b. No. Because for \(2x=54 - 3y\), since \(2x\) is even and 54 is even, \(3y\) must be even, and since 3 is odd, \(y\) must be even.
c. Two possible solutions: \((3,16)\) (3 two - point baskets and 16 three - point baskets), \((6,14)\) (6 two - point baskets and 14 three - point baskets)