Answer
$y = -\frac{1}{8}x + \frac{21}{2}$
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 $y = 8x - 1$ is $8$.
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:
$8(m) = -1$
Divide each side by $8$ to solve for $m$:
$m = \frac{-1}{8}$
Now that we have the slope of the unknown line, we can plug this and the point that we are given $(4, 10)$ 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:
$y - 10 = -\frac{1}{8}(x - 4)$
Let's convert this equation into point-intercept form:
Use the distributive property:
$y - 10 = -\frac{1}{8}x - (\frac{1}{8})(-4)$
Multiply to simplify:
$y - 10 = -\frac{1}{8}x + \frac{4}{8}$
Simplify the fraction by dividing the numerator and the denominator by their greatest common factor, which is $4$:
$y - 10 = -\frac{1}{8}x + \frac{1}{2}$
To isolate $y$, add $10$ to each side of the equation:
$y = -\frac{1}{8}x + \frac{1}{2} + 10$
We need to find an equivalent fraction for $10$ so that the denominator is $2$:
$y = -\frac{1}{8}x + \frac{1}{2} + \frac{20}{2}$
Add the fractions to simplify:
$y = -\frac{1}{8}x + \frac{21}{2}$
This is the equation of the line that we are looking for in slope-intercept form.
