Line p is defined by the equation 2x + 5y = 10. Line q is perpendicular to line p and has … — ACT

Line $p$ is defined by the equation $2x + 5y = 10$ . Line $q$ is perpendicular to line $p$ and has a $y$ -intercept of $-3$ . If line $q$ passes through the point $(a, 7)$ , what is the value of $a$ ?

Answer: 4

Answer

The value of

a

is 4.

First, the equation of line

p

2x + 5y = 10

, is converted to slope-intercept form to find its slope:

y = -\frac{2}{5}x + 2

. This indicates the slope of line

p

-\frac{2}{5}

. Because line

q

is perpendicular to line

p

, its slope must be the negative reciprocal of

-\frac{2}{5}

, which is

\frac{5}{2}

. Using the given

y

-intercept of

-3

, the equation of line

q

is written as

y = \frac{5}{2}x - 3

. Substituting the coordinates of the point

(a, 7)

into this equation gives

7 = \frac{5}{2}a - 3

. Adding

3

to both sides results in

10 = \frac{5}{2}a

, and solving for

a

yields

a = 4

Step-by-Step Solution

Find the slope of line

p

The slope of line

p

-\frac{2}{5}

Rewriting the equation

2x + 5y = 10

in slope-intercept form (

y = mx + b

) gives

5y = -2x + 10

, which simplifies to

y = -\frac{2}{5}x + 2

. The slope

m

is the coefficient of

x

, which is

-\frac{2}{5}

Determine the slope of line

q

The slope of line

q

\frac{5}{2}

Since line

q

is perpendicular to line

p

, its slope is the negative reciprocal of line

p

's slope:

-\frac{1}{-\frac{2}{5}} = \frac{5}{2}

Write the equation of line

q

The equation of line

q

y = \frac{5}{2}x - 3

Line

q

has a slope of

\frac{5}{2}

and a

y

-intercept of

-3

. Using the slope-intercept form

y = mx + b

gives

y = \frac{5}{2}x - 3

Substitute the point

(a, 7)

into the equation of line

q

and solve for

a

a = 4

Substituting

x = a

and

y = 7

into the equation gives

7 = \frac{5}{2}a - 3

. Adding

3

to both sides yields

10 = \frac{5}{2}a

. Multiplying both sides by

2

gives

20 = 5a

, and dividing by

5

results in

a = 4

Key Concept

The slope of a line perpendicular to a given line is the negative reciprocal of the given line's slope.

Estimated Time:1m 30s

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