Answer
The x-intercept is $10$. The y-intercept is $-\frac{10}{3}$.
Work Step by Step
To find the x-intercept and y-intercept of the line, we first need to find the equation of the line.
We can use the two points given to formulate the point-slope form.
Let's first find the slope:
$m = \frac{y_2 - y_1}{x_2 - x_1}$, where $m$ is the slope and $(x_1, y_1)$ and $(x_2, y_2)$ are points on the line.
Let's plug in our two points into this formula:
$m = \frac{-2 - (-5)}{4 - (-5)}$
Simplify by adding or subtracting in the numerator and denominator:
$m = \frac{3}{9}$
Simplify by dividing both the numerator and denominator by their greatest common factor, $3$:
$m = \frac{1}{3}$
Since we have the slope and two points, we can use the point-slope form, which is given by the following formula:
$y - y_1 = m(x - x_1)$
Let's plug in the slope and a point into this formula:
$y - (-2) = \frac{1}{3}(x - 4)$
Simplify the right side of the equation:
$y + 2 = \frac{1}{3}(x - 4)$
This is the point-slope formula of the equation.
To find the x-intercept, we set $y$ equal to $0$:
$0 + 2 = \frac{1}{3}(x - 4)$
Use the distributive property on the right side of the equation:
$2 = \frac{1}{3}x - \frac{4}{3}$
Add $\frac{4}{3}$ to both sides of the equation to move constants to one side of the equation:
$\frac{1}{3}x = 2 + \frac{4}{3}$
Add the fractions on the right side of the equation:
$\frac{1}{3}x = \frac{10}{3}$
Divide each side by $\frac{1}{3}$ to solve for $x$:
$x = \frac{30}{3}$
Divide both the numerator and denominator by their greatest common factor, $3$:
$x = 10$
To find the y-intercept, we set $x$ equal to $0$:
$y + 2 = \frac{1}{3}(0 - 4)$
Evaluate parentheses first:
$y + 2 = \frac{1}{3}(-4)$
Multiply on the right side of the equation:
$y + 2 = -\frac{4}{3}$
Subtract $2$ from each side of the equation:
$y = -\frac{4}{3} - 2$
Subtract to solve for $y$:
$y = -\frac{10}{3}$
The x-intercept is $10$. The y-intercept is $-\frac{10}{3}$.
