To determine the [tex]$y$[/tex]-intercept of a line that has a slope of -3 and passes through the point [tex]$(-5, 4)$[/tex], we can use the point-slope form of the equation of a line.
The point-slope form of a line is given by:
[tex]\[
y = mx + b
\][/tex]
where [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the [tex]$y$[/tex]-intercept.
Given:
- The slope [tex]\( m = -3 \)[/tex]
- The point [tex]\((x_1, y_1) = (-5, 4)\)[/tex]
We need to find the [tex]$y$[/tex]-intercept [tex]\( b \)[/tex].
Substitute the slope and the coordinates of the point into the equation:
[tex]\[
y_1 = m \cdot x_1 + b
\][/tex]
[tex]\[
4 = -3 \cdot (-5) + b
\][/tex]
First, multiply the slope [tex]\(-3\)[/tex] by [tex]\( x_1 = -5 \)[/tex]:
[tex]\[
-3 \cdot (-5) = 15
\][/tex]
Then, substitute this value back into the equation:
[tex]\[
4 = 15 + b
\][/tex]
To isolate [tex]\( b \)[/tex], subtract 15 from both sides of the equation:
[tex]\[
4 - 15 = b
\][/tex]
[tex]\[
b = -11
\][/tex]
Therefore, the [tex]$y$[/tex]-intercept of the line is [tex]\( -11 \)[/tex].
So the correct answer is:
[tex]\[
\boxed{-11}
\][/tex]
