Certainly! Let's work through the problem step-by-step:
1. Start with the given equation of the line:
[tex]\[
-x + 4y = 32
\][/tex]
2. Convert this equation to slope-intercept form (y = mx + b), where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept:
- Isolate the [tex]\( y \)[/tex] on one side of the equation.
[tex]\[
4y = x + 32
\][/tex]
- Divide each term by 4 to solve for [tex]\( y \)[/tex].
[tex]\[
y = \frac{1}{4}x + 8
\][/tex]
3. Identify the slope of the given line:
- From the equation [tex]\( y = \frac{1}{4}x + 8 \)[/tex], we see the slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{4} \)[/tex].
4. Determine the slope of the parallel line:
- Parallel lines have the same slope. Therefore, the slope of the new line is also [tex]\( \frac{1}{4} \)[/tex].
5. Use the point-slope form of the equation to find the y-intercept of the new line:
- The point-slope form equation 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.
- We are given that the line passes through the point [tex]\((-4, -4)\)[/tex]. So, [tex]\( x_1 = -4 \)[/tex] and [tex]\( y_1 = -4 \)[/tex], and our slope [tex]\( m = \frac{1}{4} \)[/tex].
- Substitute these values into the point-slope form equation:
[tex]\[
y - (-4) = \frac{1}{4}(x - (-4))
\][/tex]
[tex]\[
y + 4 = \frac{1}{4}(x + 4)
\][/tex]
6. Solve for [tex]\( y \)[/tex] to convert to slope-intercept form:
- Distribute the [tex]\( \frac{1}{4} \)[/tex] in the equation.
[tex]\[
y + 4 = \frac{1}{4}x + 1
\][/tex]
- Subtract 4 from both sides to isolate [tex]\( y \)[/tex].
[tex]\[
y = \frac{1}{4}x + 1 - 4
\][/tex]
[tex]\[
y = \frac{1}{4}x - 3
\][/tex]
7. Write the final equation:
- After simplifying, the equation of the line in slope-intercept form is:
[tex]\[
y = \frac{1}{4}x - 3
\][/tex]
Hence, the equation of the line parallel to [tex]\(-x + 4y = 32\)[/tex] and passing through the point [tex]\((-4, -4)\)[/tex] is:
[tex]\[
y = \frac{1}{4}x - 3
\][/tex]
