In the standard (x, y) coordinate plane, the line L_1 is parallel to the line y = 3x - 7. … — ACT

In the standard $(x, y)$ coordinate plane, the line $L_1$ is parallel to the line $y = 3x - 7$ . The line $L_2$ is perpendicular to $L_1$ and passes through the point $(6, 5)$ . If the $y$ -intercept of $L_2$ is $(0, c)$ , what is the value of $c$ ?

Answer: 7

Answer

The value of

c

7

Since the line

L_1

is parallel to

y = 3x - 7

, its slope is

3

. The line

L_2

is perpendicular to

L_1

, so its slope is the negative reciprocal of

3

, which is

-\frac{1}{3}

. Using the point-slope formula with the point

(6, 5)

yields the equation of

L_2

y - 5 = -\frac{1}{3}(x - 6)

, which simplifies to

y = -\frac{1}{3}x + 7

. The

y

-intercept of this line is

(0, 7)

, which gives

c = 7

Step-by-Step Solution

Determine the slope of line

L_1

Slope of

L_1

3

Parallel lines have equal slopes, and the given reference line

y = 3x - 7

has a slope of

3

Determine the slope of line

L_2

Slope of

L_2

-\frac{1}{3}

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

Find the equation of line

L_2

The equation is

y = -\frac{1}{3}x + 7

Use the point-slope form

y - y_1 = m(x - x_1)

with point

(6, 5)

and slope

-\frac{1}{3}

Identify the

y

-intercept constant

c

c = 7

The equation of

L_2

is in slope-intercept form

y = mx + b

, meaning the

y

-intercept is

(0, 7)

Key Concept

The relationship between the slopes of parallel lines (which are equal) and perpendicular lines (which are negative reciprocals).

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