Point OnePrep

SAT Math: Points on a Dependent Linear System’s Graph

A hard Digital SAT Algebra question. The system 5x + 8y = 9 and 15x + 24y = 27 is dependent — find the parametric form that gives every point on its graph.

Question

5x+8y=95x + 8y = 9
15x+24y=2715x + 24y = 27

For each real number rr, which of the following points lies on the graph of each equation in the xyxy-plane for the given system?

Step-by-Step Solution

Recognize the dependent system first — then a parametric form falls out in one step.

1Notice the two equations describe the same line.

Multiplying the first equation by 3 gives the second exactly:

3 (5x+8y)=3 (9)  ⟹  15x+24y=273\,(5x + 8y) = 3\,(9) \;\Longrightarrow\; 15x + 24y = 27

So both equations have the same solution set. “Lies on the graph of each equation” just means “lies on this one line.” The question is asking for a parametric form of every point on that line.

2Solve for y in terms of x.

Use the simpler equation:

5x+8y=9  ⟹  8y=−5x+9  ⟹  y=−5x8+985x + 8y = 9 \;\Longrightarrow\; 8y = -5x + 9 \;\Longrightarrow\; y = -\dfrac{5x}{8} + \dfrac{9}{8}

Every point on the line is determined by its x-coordinate; the y-coordinate is forced.

3Parameterize with r.

Let x = r. Then y = −5r/8 + 9/8, so every point on the line has the form:

( r,  −5r8+98 )\left(\,r,\; -\dfrac{5r}{8} + \dfrac{9}{8}\,\right)

That is option A exactly.

4Sanity-check by plugging r = 0 into all four choices.

At r = 0 the choices give:

A → (0, 9/8). Check 5x + 8y: 5(0) + 8(9/8) = 9. ✓

B → (9/8, 0). Check: 5(9/8) + 8(0) = 45/8 ≠ 9. ✗

C → (9, 27). Check: 5(9) + 8(27) = 45 + 216 = 261 ≠ 9. ✗

D → (9, 27). Same failure as C. ✗

Only A satisfies the equation at r = 0, and the algebra above shows it works for every r. The answer is A.

5Calculator path: Desmos with a slider.

This question is built for a graphing approach. In Desmos:

1) Type both equations: 5x + 8y = 9 and 15x + 24y = 27. They draw a single line — the system is dependent.

2) Type each answer choice as a point. For A, enter (r, -5r/8 + 9/8). Desmos prompts you to add a slider for r — accept it.

3) Drag the slider for r. Watch which point stays on the line for every value of r.

Only A traces along the line as r varies. B sits off the line entirely (it only hits the line for one specific r), and C and D land at points like (9, 27) that are nowhere near it.

This is the fastest reliable solution and a great template for any SAT problem that asks “which of the following lies on the graph for all r/t/k.”