Find the x and z intercepts of the graph of 3x - z $\leq$ 9

Analyze the question: We need the intercepts of the graph of 3x - z ≤ 9. The boundary line is 3x - z = 9 (the "≤" only tells us which side is shaded). The "x and z intercepts" means where the line crosses the x-axis and the z-axis (the z variable plays the role of y).

Concept:

• The x-intercept is found by setting z = 0.
• The z-intercept is found by setting x = 0.

Solution:

x-intercept (z = 0): 3x - 0 = 9, so x = 3. The point is (3, 0).

z-intercept (x = 0): 0 - z = 9, so z = -9. The point is (0, -9).

So the intercepts are x = 3 and z = -9.

Matching the options:

• A. (3, -9): this lists the x-intercept value 3 and the z-intercept value -9 together, which matches our results.
• B. (-3, 9): wrong signs.
• C. (-3, -9): the x-intercept sign is wrong.
• D. (-3, 0): wrong.

Option A is not a single point on the line. Check: 3(3) - (-9) = 18, which is not 9. So it is best read as the pair (x-intercept, z-intercept) = (3, -9), not as a coordinate point.

Answer: A

Tip: For intercepts, cover up one variable at a time. Set one variable to 0 to find the intercept on the other axis. When an option looks like a "point" but doesn't lie on the line, check whether the question is asking you to list the two intercept values instead.