Three straight lines, $l_{1}, l_{2}$ and $l_{3}$, have slopes $\frac{1}{2}, \frac{1}{3}$ and $\frac{1}{4}$, respectively. All three lines have the same $y$-intercept. If the sum of the $x$-intercepts of the three lines is 36 , then the $y$-intercept is
(A) $\frac{-13}{12}$
(B) $\frac{-12}{13}$
(C) -4
(D) 4
(E) -9
Three straight lines, $l_{1}, l_{2}$ and $l_{3}$, have slopes $\frac{1}{2}, \frac{1}{3}$ and $\frac{1}{4}$, respectively. All three lines have the same $y$-intercept. If the sum of the $x$-intercepts of the three lines is 36 , then the $y$-intercept is
(A) $\frac{-13}{12}$
(B) $\frac{-12}{13}$
(C) -4
(D) 4
(E) -9

## Solution

Let $b$ represent the common $y$-intercept of the three lines. The first line, $l_{1}$, has equation $y=\frac{1}{2} x+b$. Set $y=0$ in this equation to find the $x$-intercept of the first line.

$$
\begin{aligned}
0 & =\frac{1}{2} x+b \\
-\frac{1}{2} x & =b \\
x & =-2 b
\end{aligned}
$$

Similarly the second line, $l_{2}$, has equation $y=\frac{1}{3} x+b$ and $x$-intercept $-3 b$. The third line, $l_{3}$, has equation $y=\frac{1}{4} x+b$ and $x$-intercept $-4 b$.

We know that $-2 b-3 b-4 b=36$

$$
\begin{aligned}
-9 b & =36 \\
b & =-4
\end{aligned}
$$

Thus, the common $y$-intercept of the three lines is -4 .

ANSWER: (C)