In the xy-plane, line L passes through the points (0, 4) and (6, 0). A second line, M, is … — SAT

In the $xy$ -plane, line $L$ passes through the points $(0, 4)$ and $(6, 0)$ . A second line, $M$ , is perpendicular to line $L$ and intersects line $L$ at a point on the line $y = x$ . If line $M$ is represented by the equation $y = px + q$ , where $p$ and $q$ are constants, what is the value of $p + q$ ?

A
$\frac{6}{5}$
B
$\frac{10}{3}$
$\frac{3}{10}$ Answer
D
$-\frac{9}{2}$

Answer

The correct answer is the value stating three-tenths

To find the sum of the slope

p

and

y

-intercept

q

of line

M

, we first determine the equation of line

L

. Line

L

passes through

(0, 4)

and

(6, 0)

, so its slope is

\frac{0 - 4}{6 - 0} = -\frac{2}{3}

. The equation of line

L

is therefore

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

. Setting

y = x

gives

x = -\frac{2}{3}x + 4

, which yields

x = 2.4

. The intersection point is

(2.4, 2.4)

. Because line

M

is perpendicular to line

L

, its slope is the negative reciprocal of

-\frac{2}{3}

, which is

p = 1.5

. Using point-slope form with the intersection point

(2.4, 2.4)

, the equation of line

M

y - 2.4 = 1.5(x - 2.4)

, which simplifies to

y = 1.5x - 1.2

. Thus,

q = -1.2

. The sum

p + q

1.5 - 1.2 = 0.3

, or

\frac{3}{10}

Step-by-Step Solution

Find the equation of line

L

using the given points

(0, 4)

and

(6, 0)

The slope of line

L

m_L = \frac{0 - 4}{6 - 0} = -\frac{2}{3}

. Since the

y

-intercept is

(0, 4)

, the equation of line

L

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

To find where line

L

intersects another line, we first need to establish its linear equation.

Find the intersection point of line

L

and the line

y = x

Substitute

y = x

into the equation for line

L

x = -\frac{2}{3}x + 4 \implies \frac{5}{3}x = 4 \implies x = 2.4

. Thus, the intersection point is

(2.4, 2.4)

The problem states that line

M

intersects line

L

at a point on the line

y = x

Determine the equation of line

M

using the intersection point and the perpendicular relationship.

Since line

M

is perpendicular to line

L

, its slope

p

is the negative reciprocal of

-\frac{2}{3}

, which is

p = \frac{3}{2} = 1.5

. Using point-slope form at

(2.4, 2.4)

y - 2.4 = 1.5(x - 2.4) \implies y = 1.5x - 1.2

. This gives

q = -1.2

Perpendicular lines in the coordinate plane have slopes that multiply to negative one. We then use the point-slope formula to define the equation of line

M

Calculate the sum of

p

and

q

p + q = 1.5 + (-1.2) = 0.3 = \frac{3}{10}

The question asks for the value of the expression

p + q

Key Concept

Perpendicular lines and linear graph intersections

Estimated Time:3m 0s

Question

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Key Concept

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