To find the equation of the line in slope-intercept form, [tex]\( y = mx + b \)[/tex], we need to determine the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex].
1. We are given the slope [tex]\( m = -\frac{3}{2} \)[/tex] and a point [tex]\((-10, 8)\)[/tex] that the line passes through.
2. We will use the slope-intercept form [tex]\( y = mx + b \)[/tex] and substitute the given point and slope into this equation to solve for [tex]\( b \)[/tex], the y-intercept.
3. Start with the equation:
[tex]\[
y = mx + b
\][/tex]
4. Substitute [tex]\( x = -10 \)[/tex], [tex]\( y = 8 \)[/tex], and [tex]\( m = -\frac{3}{2} \)[/tex]:
[tex]\[
8 = -\frac{3}{2} \cdot (-10) + b
\][/tex]
5. Simplify the multiplication:
[tex]\[
8 = 15 + b
\][/tex]
6. Solve for [tex]\( b \)[/tex] by isolating [tex]\( b \)[/tex] on one side of the equation:
[tex]\[
b = 8 - 15
\][/tex]
7. Simplify the subtraction:
[tex]\[
b = -7
\][/tex]
8. Now that we have the y-intercept [tex]\( b = -7 \)[/tex], we can write the equation of the line in slope-intercept form by substituting [tex]\( m = -\frac{3}{2} \)[/tex] and [tex]\( b = -7 \)[/tex] into the form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y = -\frac{3}{2}x - 7
\][/tex]
Thus, the equation of the line in slope-intercept form is:
[tex]\[
y = -\frac{3}{2}x - 7
\][/tex]
