In the xy-plane, line L_1 is defined by the equation ax + by = c, where a, b, and c are no… — SAT

In the $xy$ -plane, line $L_1$ is defined by the equation $ax + by = c$ , where $a$ , $b$ , and $c$ are non-zero constants. Line $L_2$ is perpendicular to line $L_1$ and has a $y$ -intercept that is twice the $y$ -intercept of line $L_1$ . If the $x$ -intercept of line $L_2$ is $d$ , which of the following expressions represents $d$ in terms of $a$ , $b$ , and $c$ ?

$-\frac{2ac}{b^2}$ Answer
B
$\frac{2ac}{b^2}$
C
$-\frac{2bc}{a^2}$
D
$-\frac{2c}{a}$

Answer

The expression that represents

d

in terms of

a

b

, and

c

-\frac{2ac}{b^2}

The correct answer is found by first rewriting the equation of the first line in slope-intercept form to determine its slope,

-\frac{a}{b}

, and its

y

-intercept,

\frac{c}{b}

. The second line is perpendicular to the first, so its slope is the negative reciprocal,

\frac{b}{a}

. Its

y

-intercept is twice that of the first line, which is

\frac{2c}{b}

. Setting

y = 0

in the equation of the second line,

y = \frac{b}{a}x + \frac{2c}{b}

, and solving for

x

yields the

x

-intercept

d = -\frac{2ac}{b^2}

Step-by-Step Solution

Rewrite the equation of line

L_1

in slope-intercept form.

y = -\frac{a}{b}x + \frac{c}{b}

To identify the slope and

y

-intercept of the first line.

Determine the slope and

y

-intercept of line

L_2

Slope of

L_2

\frac{b}{a}

;

y

-intercept of

L_2

\frac{2c}{b}

Line

L_2

is perpendicular to

L_1

(so its slope is the negative reciprocal of

-\frac{a}{b}

) and its

y

-intercept is twice that of

L_1

Write the equation of line

L_2

y = \frac{b}{a}x + \frac{2c}{b}

Using the slope-intercept form

y = mx + k

for the second line.

Find the

x

-intercept

d

of line

L_2

by setting

y = 0

d = -\frac{2ac}{b^2}

The

x

-intercept is the value of

x

when

y = 0

Key Concept

Linear Equations in Two Variables

Alternative Method

Instead of converting to slope-intercept form, one can use the general form relationships. For

ax + by = c

, the slope is

-a/b

and the

y

-intercept is

c/b

. A perpendicular line has slope

b/a

. With double the

y

-intercept, its equation can be written as

y - \frac{2c}{b} = \frac{b}{a}x

. Setting

y=0

directly gives

-\frac{2c}{b} = \frac{b}{a}d \implies d = -\frac{2ac}{b^2}

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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