2006 iTest Problems/Problem 5

Problem

A line has y-intercept $(0,3)$ and forms a right angle to the line $2x + y = 3$ . Find the x-intercept of the line.

$\mathrm{(A)}\,(4,0)\quad\mathrm{(B)}\,(6,0)\quad\mathrm{(C)}\,(-4,0)\quad\mathrm{(D)}\,(-6,0)\quad\mathrm{(E)}\,\text{none of the above}$

Solution

The given line's equation is $y = -2x + 3$ . Since the wanted line forms a right angle to the given line, the slopes of the two lines multiply to $-1$ , so the slope of the wanted line is $\tfrac12$ . Thus, the equation of the wanted line is $y = \tfrac12 x + 3$ .

To find the x-intercept of the line, substitute $0$ for $y$ and solve for $x$ . $\begin{align*} \tfrac12 x + 3 &= 0 \\ \tfrac12 x &= -3 \\ x &= -6 \end{align*}$ The x-intercept of that line is $\boxed{\textbf{(D)}\,(-6,0)}$ .

2006 iTest (Problems, Answer Key)
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2006 iTest Problems/Problem 5

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