Digital SAT Math: Where Does the Line Through (2, 5) and (6, 13) Cross the x-Axis?

Exact Question Text

A line in the $xy$-plane passes through the points $(2, 5)$ and $(6, 13)$. What is the $x$-coordinate of the point where the line crosses the $x$-axis?

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A) $-\frac{1}{2}$ B) $\frac{1}{2}$ C) $-1$ D) $1$

1. Step-by-Step Formal Solution

Slope:

$$ m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2 $$

Equation (point-slope with $(2, 5)$):

$$ y - 5 = 2(x - 2) \implies y = 2x + 1 $$

Check with the second point: $2(6) + 1 = 13$ ✓

x-intercept: set $y = 0$.

$$ 0 = 2x + 1 \implies x = -\frac{1}{2} $$

Answer: A

2. Fast Desmos Shortcut

Enter y = 2x + 1 (or type both points and use the regression line y ~ mx + b). Click the spot where the line meets the horizontal axis. Desmos shows $(-0.5, 0)$. Reading the $x$-value gives $-\frac{1}{2}$.

3. Distractor Trap Analysis

• D) 1: this is the y-intercept of the line ($y = 1$ when $x = 0$). The question asks for the $x$-coordinate where the line crosses the $x$-axis.
• B) 1/2: a sign error when solving $2x + 1 = 0$. Subtract 1 first: $2x = -1$.
• C) -1: forgetting to divide by the slope: $2x = -1$ gives $x = -\frac{1}{2}$, not $-1$.

4. The General Pattern

For a line $y = mx + b$, the $x$-intercept is $-\frac{b}{m}$ and the $y$-intercept is $b$. Here $-\frac{b}{m} = -\frac{1}{2}$.

5. Two More to Practice (Same Method)

1. Line through $(1, 7)$ and $(4, 1)$: slope $= \frac{1 - 7}{4 - 1} = -2$, so $y = -2x + 9$, and the $x$-intercept is $x = \frac{9}{2}$.
2. Line through $(-3, 0)$ and $(0, 6)$: slope $= 2$, $y = 2x + 6$. The $x$-intercept is the given point, $x = -3$.

For another SAT linear-equations setup, see the system with infinitely many solutions.