Answer
$y = -\frac{2}{5}x + \frac{29}{5}$
Work Step by Step
We are given the points $(2, 5)$ and $(12, 1)$.
Let's use the formula to find the slope $m$ given two points:
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Let's plug in the values into this formula:
$m = \frac{1 - 5}{12 - 2}$
Subtract the numerator and denominator to simplify:
$m = \frac{-4}{10}$
Divide the numerator and denominator by their greatest common denominator, which is $2$:
$m = -\frac{2}{5}$
Now that we have the slope, we can use one of the points and plug these values into the point-slope equation, which is given by the formula:
$y - y_1 = m(x - x_1)$
Let's plug in the points and slope into the formula:
$y - 5 = -\frac{2}{5}(x - 2)$
This equation is now in point-slope form. To change this equation into the point-intercept form, we need to isolate $y$.
Use distribution to simplify:
$y - 5 = -\frac{2}{5}x - \frac{2}{5}(-2)$
Simplify by multiplying:
$y - 5 = -\frac{2}{5}x + \frac{4}{5}$
To isolate $y$, we add $5$ to each side of the equation:
$y = -\frac{2}{5}x + \frac{4}{5} + 5$
Change $5$ into an equivalent fraction that has $5$ as its denominator so that both fractions have the same denominator:
$y = -\frac{2}{5}x + \frac{4}{5} + \frac{25}{5}$
Add the fractions to simplify:
$y = -\frac{2}{5}x + \frac{29}{5}$
Now, we have the equation of the line in slope-intercept form.
