Answer
$y = \frac{1}{3}x + 4$
Work Step by Step
For perpendicular lines, the product of the slopes of the two lines is $-1$.
The equation of the line that we are given is in slope-intercept form, which is given by the formula:
$y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
The slope of the line given is $-3$.
The product of the slopes of the two lines must equal $-1$, so let us set up the equation to find the slope $m$ of the unknown line:
$-3(m) = -1$
Divide each side by $-3$ to solve for $m$:
$m = \frac{-1}{-3}$
Divide both the numerator and denominator by $-1$ to simplify the fraction:
$m = \frac{1}{3}$
Now that we have the slope of the unknown line, we can plug this and the point that we are given $(3, 5)$ into point-slope form, which is given by the formula:
$y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a point on that graph.
Let us plug in the values:
$y - 5 = \frac{1}{3}(x - 3)$
Let's convert this equation into point-intercept form:
Use the distributive property:
$y - 5 = \frac{1}{3}x + \frac{1}{3}(-3)$
Multiply to simplify:
$y - 5 = \frac{1}{3}x - \frac{3}{3}$
Simplify the fraction by dividing the numerator by the denominator:
$y - 5 = \frac{1}{3}x - 1$
To isolate $y$, add $5$ to each side of the equation:
$y = \frac{1}{3}x - 1 + 5$
Add to simplify:
$y = \frac{1}{3}x + 4$
This is the equation of the line that we are looking for in slope-intercept form.
