In the xy-plane, line L is given by the equation 3x + 2y = 18. Line N is perpendicular to … — GRE General Test

In the $xy$ -plane, line $L$ is given by the equation $3x + 2y = 18$ . Line $N$ is perpendicular to line $L$ and passes through the point $(-1, 4)$ . If line $L$ and line $N$ intersect at the point $(p, q)$ , what is the value of $p + q$ ?

Answer: 8

Answer

Converting the given equation

3x + 2y = 18

into slope-intercept form yields

y = -\frac{3}{2}x + 9

, so line

L

has slope

-\frac{3}{2}

. A perpendicular line must have a slope equal to the negative reciprocal, which is

\frac{2}{3}

. Using point

(-1, 4)

in the point-slope form gives

y - 4 = \frac{2}{3}(x + 1)

, simplifying to

2x - 3y = -14

. Solving the system of equations formed by line

L

(

3x + 2y = 18

) and line

N

(

2x - 3y = -14

) via elimination yields

x = 2

and

y = 6

. Therefore, the intersection point is

(2, 6)

, and

p + q = 2 + 6 = 8

Step-by-Step Solution

Determine the slope of line

L

Slope of line

L

m_L = -\frac{3}{2}

Convert

3x + 2y = 18

y = -\frac{3}{2}x + 9

to identify the slope coefficient of

x

Calculate the perpendicular slope for line

N

Slope of line

N

m_N = \frac{2}{3}

Perpendicular lines have slopes that are negative reciprocals of each other.

Derive the equation for line

N

Equation of line

N

2x - 3y = -14

Apply point-slope form

y - 4 = \frac{2}{3}(x + 1)

and rearrange into standard linear form.

Solve the system of equations to find the intersection point

(p, q)

p = 2

and

q = 6

, giving point

(2, 6)

Eliminate variable

y

by adding

3 \times (3x + 2y = 18)

and

2 \times (2x - 3y = -14)

to get

13x = 26

Sum the coordinates

p

and

q

p + q = 8

Evaluate

2 + 6 = 8

as required by the stem.

Key Concept

Perpendicular line slope relationships and linear system intersection

Estimated Time:2m 0s

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