Sure, let's go through the steps to find the slope-intercept form of the line that passes through the point [tex]\((-1,3)\)[/tex] and is parallel to the line [tex]\(5x + 2y = -3\)[/tex].
1. Identify the slope of the given line:
- The equation of the line is [tex]\(5x + 2y = -3\)[/tex].
- To identify the slope, we should first re-write this in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
2. Convert to slope-intercept form:
- Starting with [tex]\(5x + 2y = -3\)[/tex], we need to solve for [tex]\(y\)[/tex].
- Subtract [tex]\(5x\)[/tex] from both sides to get: [tex]\(2y = -5x - 3\)[/tex].
- Divide everything by 2: [tex]\(y = -\frac{5}{2}x - \frac{3}{2}\)[/tex].
The slope [tex]\(m\)[/tex] of the line is [tex]\(-\frac{5}{2}\)[/tex].
3. Find the slope of the parallel line:
- Since parallel lines have the same slope, the slope of the line we're looking for is also [tex]\(-\frac{5}{2}\)[/tex].
4. Use the point-slope form to find the y-intercept ([tex]\(b\)[/tex]):
- We know the slope [tex]\(m = -\frac{5}{2}\)[/tex] and the line passes through the point [tex]\((-1, 3)\)[/tex].
- The point-slope form of a line is [tex]\(y = mx + b\)[/tex].
- Substitute [tex]\(x = -1\)[/tex], [tex]\(y = 3\)[/tex], and [tex]\(m = -\frac{5}{2}\)[/tex] into the equation to find [tex]\(b\)[/tex].
[tex]\[3 = -\frac{5}{2}(-1) + b\][/tex]
5. Solve for [tex]\(b\)[/tex]:
- Calculate the right-hand side: [tex]\(-\frac{5}{2} \times (-1) = \frac{5}{2}\)[/tex].
- Therefore: [tex]\[3 = \frac{5}{2} + b\][/tex]
- Subtract [tex]\(\frac{5}{2}\)[/tex] from both sides to solve for [tex]\(b\)[/tex]:
[tex]\[b = 3 - \frac{5}{2}\][/tex]
[tex]\[b = \frac{6}{2} - \frac{5}{2}\][/tex]
[tex]\[b = \frac{1}{2}\][/tex]
6. Write the equation in slope-intercept form:
- Now we have the slope [tex]\(m = -\frac{5}{2}\)[/tex] and the y-intercept [tex]\(b = \frac{1}{2}\)[/tex], so the equation of the line is:
[tex]\[y = -\frac{5}{2}x + \frac{1}{2}\][/tex]
Thus, the slope-intercept form of the line that passes through the point [tex]\((-1, 3)\)[/tex] and is parallel to the line [tex]\(5x + 2y = -3\)[/tex] is:
[tex]\[y = -2.5x + 0.5\][/tex]
