In the xy-plane, a line has the equation y = mx + b, where m... | SAT Linear equations in two variables Question, Answer Explained

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Math · AlgebraHard

In the $xy$ -plane, a line has the equation $y = mx + b$ , where $m$ and $b$ are nonzero constants. The line passes through the point $(2, 7)$ , and the $x$ -coordinate of its $x$ -intercept is twice the $y$ -coordinate of its $y$ -intercept. What is the value of $b$ ?

$6$

$8$

$11$

$16$

Show answer and explanation

Answer: B

The $y$ -intercept is $(0, b)$ and the $x$ -intercept is $\left(-\frac{b}{m}, 0\right)$ . The condition gives $-\frac{b}{m} = 2b$ . Since $b \ne 0$ , divide both sides by $b$ : $-\frac{1}{m} = 2$ , so $m = -\frac{1}{2}$ . Then $7 = -\frac{1}{2}(2) + b$ , so $b = 8$ . Check: $y = -\frac{1}{2}x + 8$ has $x$ -intercept $16 = 2(8)$ .

Choice D, $16$ , is the $x$ -coordinate of the $x$ -intercept, which is twice $b$ , not $b$ itself.

Why the other choices are wrong

A. This uses a slope of 1/2. The line y = (1/2)x + 6 has x-intercept -12, which isn't twice 6.
C. This uses a slope of -2. The condition -b/m = 2b gives m = -1/2.
D. 16 is the x-coordinate of the x-intercept, which is twice b. The question asks for b.

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