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The equation of line $g$ is $y = \frac{8}{9}x + 2$ . Line $h$ is parallel to line $g$ and passes through $(9,4)$ . What is the equation of line $h$ ? $\newline$ Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

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Write an equation for a parallel or perpendicular line

Full solution

Q. The equation of line

g

is

y = \frac{8}{9}x + 2

. Line

h

is parallel to line

g

and passes through

(9,4)

. What is the equation of line

h

?

\newline

Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

Determine slope of line $g$ : Determine the slope of line $g$ . $\newline$ Line $g$ has the equation $y = \frac{8}{9}x + 2$ . The slope of line $g$ is the coefficient of $x$ , which is $\frac{8}{9}$ .
Find slope of line $h$ : Since line $h$ is parallel to line $g$ , it must have the same slope. $\newline$ The slope of line $h$ is therefore also $\frac{8}{9}$ .
Use point-slope form: Use the point-slope form to find the equation of line $h$ . The point-slope form of a line is $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is a point on the line. We know that line $h$ passes through the point $(9,4)$ and has a slope of $\frac{8}{9}$ .
Plug slope and point: Plug the slope and the point into the point-slope form. $\newline$ Using the point $(9,4)$ and the slope $\frac{8}{9}$ , the equation becomes $y - 4 = \frac{8}{9}(x - 9)$ .
Simplify equation: Simplify the equation to get it into slope-intercept form, $y = mx + b$ .
$y - 4 = \frac{8}{9}(x - 9)$
$y - 4 = \frac{8}{9}x - 8$
$y = \frac{8}{9}x - 8 + 4$
$y = \frac{8}{9}x - 4$
Find y-intercept: Combine like terms to find the y-intercept $b$ . $-8 + 4 = -4$ So the y-intercept is $-4$ .
Write final equation: Write the final equation of line $h$ in slope-intercept form. $\newline$ The equation of line $h$ is $y = \frac{8}{9}x - 4$ .

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