Question

consider the following system of equations given in slope - intercept form.
$y = -\frac{1}{3}x + 17$,
$y = 5x - 23$
use the graphing calculator to determine the best window range to find the point of intersection.
$\boldsymbol{-10 \leq x \leq 10, -10 \leq y \leq 10}$
$\boldsymbol{-20 \leq x \leq 0, -20 \leq y \leq 0}$
$\boldsymbol{0 \leq x \leq 20, 0 \leq y \leq 20}$
$\boldsymbol{20 \leq x \leq 40, 20 \leq y \leq 40}$

Explanation:

Step1: Analyze the equations

We have two linear equations: \( y = -\frac{1}{3}x + 17 \) and \( y = 5x - 23 \). To find the intersection, we can also solve them algebraically first. Set the two equations equal:
\( -\frac{1}{3}x + 17 = 5x - 23 \)
Multiply through by 3 to eliminate the fraction:
\( -x + 51 = 15x - 69 \)
\( 51 + 69 = 15x + x \)
\( 120 = 16x \)
\( x = 7.5 \)
Then substitute \( x = 7.5 \) into \( y = 5x - 23 \):
\( y = 5(7.5) - 23 = 37.5 - 23 = 14.5 \)
So the intersection point is \( (7.5, 14.5) \).

Step2: Evaluate each window option

• Option 1: \( -10 \leq x \leq 10, -10 \leq y \leq 10 \). The x - value 7.5 is in this range, but the y - value 14.5 is not (since 14.5>10). So this window is too small for y.
• Option 2: \( -20 \leq x \leq 0, -20 \leq y \leq 0 \). The x - value 7.5 is positive, so it's not in this range. Eliminate this option.
• Option 3: \( 0 \leq x \leq 20, 0 \leq y \leq 20 \). The x - value 7.5 is in \( [0,20] \) and the y - value 14.5 is in \( [0,20] \). This window can show the intersection point.
• Option 4: \( 20 \leq x \leq 40, 20 \leq y \leq 40 \). The x - value 7.5 is less than 20, so it's not in this range. Eliminate this option.

Answer:

C. \( 0 \leq x \leq 20, 0 \leq y \leq 20 \) (assuming the options are labeled A, B, C, D with C being \( 0 \leq x \leq 20, 0 \leq y \leq 20 \))