Questions: If one point on a line is (3,-5) and the line's slope is -9, find the y-intercept. The y-intercept is . (Type an integer or a simplified fraction.)

$If one point on a line is (3,-5) and the line's slope is -9, find the y-intercept. The y-intercept is . (Type an integer or a simplified fraction.)$

Transcript text: If one point on a line is $(3,-5)$ and the line's slope is -9 , find the $y$-intercept. The $y$-intercept is $\square$ . (Type an integer or a simplified fraction.)

Solution

Solution Steps

Step 1: Use the point-slope form of a line

The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \] where $(x_1, y_1)$ is a point on the line and $m$ is the slope. Here, the point $(3, -5)$ lies on the line, and the slope $m = -9$. Substituting these values into the equation: \[ y - (-5) = -9(x - 3) \] Simplify the equation: \[ y + 5 = -9x + 27 \]

Step 2: Solve for $y$ to find the slope-intercept form

The slope-intercept form of a line is: \[ y = mx + b \] where $b$ is the $y$-intercept. Rearrange the equation from Step 1 to solve for $y$: \[ y = -9x + 27 - 5 \] \[ y = -9x + 22 \]

Step 3: Identify the $y$-intercept

From the slope-intercept form $y = -9x + 22$, the $y$-intercept $b$ is the constant term: \[ b = 22 \]