To identify an equation in slope-intercept form for the line parallel to [tex]\( y = 4x - 9 \)[/tex] that passes through the point [tex]\((-5, 3)\)[/tex], follow these steps:
1. Identifying the Slope of the Parallel Line:
Lines that are parallel have the same slope. The slope of the given line [tex]\( y = 4x - 9 \)[/tex] is [tex]\( 4 \)[/tex]. Thus, the slope of our new line will also be [tex]\( 4 \)[/tex].
2. Using the Point-Slope Form of the Equation:
The point-slope form of a line's equation is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\( m \)[/tex] is the slope, and [tex]\((x_1, y_1)\)[/tex] is the point through which the line passes.
In this case, [tex]\( m = 4 \)[/tex], [tex]\( x_1 = -5 \)[/tex], and [tex]\( y_1 = 3 \)[/tex].
3. Substituting the Values into the Point-Slope Form:
[tex]\[
y - 3 = 4(x - (-5))
\][/tex]
4. Simplifying the Equation:
[tex]\[
y - 3 = 4(x + 5)
\][/tex]
Distribute the [tex]\( 4 \)[/tex]:
[tex]\[
y - 3 = 4x + 20
\][/tex]
Isolate [tex]\( y \)[/tex] by adding [tex]\( 3 \)[/tex] to both sides:
[tex]\[
y = 4x + 23
\][/tex]
Therefore, the equation in slope-intercept form for the line parallel to [tex]\( y = 4x - 9 \)[/tex] that passes through [tex]\((-5, 3)\)[/tex] is:
[tex]\[ \boxed{y = 4x + 23} \][/tex]
From the given choices, the correct answer is:
C. [tex]\( y = 4x + 23 \)[/tex]
