To find the equation of the line in slope-intercept form [tex]\( y = mx + b \)[/tex] that passes through the point [tex]\((-9, 4)\)[/tex] and is parallel to the line [tex]\( 5x + 2y = -39 \)[/tex], follow these steps:
1. Find the slope of the given line:
- The given line is [tex]\( 5x + 2y = -39 \)[/tex].
- Convert this to slope-intercept form ([tex]\( y = mx + b \)[/tex]) to find the slope [tex]\( m \)[/tex].
- Solve for [tex]\( y \)[/tex]:
[tex]\[
2y = -5x - 39
\][/tex]
[tex]\[
y = -\frac{5}{2}x - \frac{39}{2}
\][/tex]
- The slope ([tex]\( m \)[/tex]) of the given line is [tex]\( -\frac{5}{2} \)[/tex].
2. Determine the slope of the parallel line:
- Lines that are parallel have the same slope.
- Therefore, the slope of the line we are looking for is also [tex]\( -\frac{5}{2} \)[/tex].
3. Use the slope-point form of the line equation:
- The slope-point form is given by [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\((x_1, y_1)\)[/tex] is a point on the line.
- Here, the line passes through the point [tex]\((-9, 4)\)[/tex].
- Substitute [tex]\( m = -\frac{5}{2} \)[/tex], [tex]\( x_1 = -9 \)[/tex], and [tex]\( y_1 = 4 \)[/tex] into the equation:
[tex]\[
y - 4 = -\frac{5}{2}(x + 9)
\][/tex]
4. Simplify to slope-intercept form:
- Distribute the slope on the right-hand side:
[tex]\[
y - 4 = -\frac{5}{2}x - \frac{5}{2} \cdot 9
\][/tex]
[tex]\[
y - 4 = -\frac{5}{2}x - \frac{45}{2}
\][/tex]
- Add 4 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{5}{2}x - \frac{45}{2} + 4
\][/tex]
- Convert 4 to a fraction with a common denominator:
[tex]\[
4 = \frac{8}{2}
\][/tex]
[tex]\[
y = -\frac{5}{2}x - \frac{45}{2} + \frac{8}{2}
\][/tex]
- Combine the constants:
[tex]\[
y = -\frac{5}{2}x - \frac{45 + 8}{2}
\][/tex]
[tex]\[
y = -\frac{5}{2}x - \frac{37}{2}
\][/tex]
So, the equation of the line in slope-intercept form is:
[tex]\[ y = -\frac{5}{2}x - \frac{37}{2} \][/tex]
From this process, we see that the correct option is:
[tex]\[ A: \quad y = -\frac{5}{2}x - \frac{37}{2} \][/tex]
