Question
Use the given conditions to write an equation for the line in point-slope form and general form.
Passing through left parenthesis negative 6 comma 9 right parenthesis and parallel to the line whose equation is 5 x minus 8 y minus 3 equals 0
Question content area bottom
Part 1
The equation of the line in point-slope form is
enter your response here.
(Type an equation. Use integers or fractions for any numbers in the equation.)
Passing through left parenthesis negative 6 comma 9 right parenthesis and parallel to the line whose equation is 5 x minus 8 y minus 3 equals 0
Question content area bottom
Part 1
The equation of the line in point-slope form is
enter your response here.
(Type an equation. Use integers or fractions for any numbers in the equation.)
Answers
Answered by
GPT-4o mini
To begin, we need to find the slope of the line given by the equation \(5x - 8y - 3 = 0\). We can rewrite this equation in slope-intercept form \(y = mx + b\) to identify the slope \(m\).
Starting with the given equation:
\[
5x - 8y - 3 = 0
\]
we can rearrange it to solve for \(y\):
\[
-8y = -5x + 3
\]
Dividing every term by -8 gives:
\[
y = \frac{5}{8}x - \frac{3}{8}
\]
From this, we see that the slope \(m\) of the original line is \(\frac{5}{8}\). Since we want to find the equation of a line that is parallel to this one, it will have the same slope: \(\frac{5}{8}\).
Now, we will use the point-slope form of the equation of a line, which is given by:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. For this problem, the point is \((-6, 9)\) and the slope \(m\) is \(\frac{5}{8}\).
Substituting the values into the point-slope form:
\[
y - 9 = \frac{5}{8}(x - (-6))
\]
Simplifying this gives:
\[
y - 9 = \frac{5}{8}(x + 6)
\]
This is the equation in point-slope form.
Now, let's write it in general form. We start from the point-slope form and manipulate it:
\[
y - 9 = \frac{5}{8}(x + 6)
\]
Multiply both sides by 8 to eliminate the fraction:
\[
8(y - 9) = 5(x + 6)
\]
Distributing both sides results in:
\[
8y - 72 = 5x + 30
\]
Rearranging this gives:
\[
-5x + 8y - 72 - 30 = 0
\]
or
\[
-5x + 8y - 102 = 0
\]
Multiplying through by -1 to make the leading coefficient positive gives the general form:
\[
5x - 8y + 102 = 0
\]
### Final Answers:
**Point-Slope Form:**
\[
y - 9 = \frac{5}{8}(x + 6)
\]
**General Form:**
\[
5x - 8y + 102 = 0
\]
Starting with the given equation:
\[
5x - 8y - 3 = 0
\]
we can rearrange it to solve for \(y\):
\[
-8y = -5x + 3
\]
Dividing every term by -8 gives:
\[
y = \frac{5}{8}x - \frac{3}{8}
\]
From this, we see that the slope \(m\) of the original line is \(\frac{5}{8}\). Since we want to find the equation of a line that is parallel to this one, it will have the same slope: \(\frac{5}{8}\).
Now, we will use the point-slope form of the equation of a line, which is given by:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. For this problem, the point is \((-6, 9)\) and the slope \(m\) is \(\frac{5}{8}\).
Substituting the values into the point-slope form:
\[
y - 9 = \frac{5}{8}(x - (-6))
\]
Simplifying this gives:
\[
y - 9 = \frac{5}{8}(x + 6)
\]
This is the equation in point-slope form.
Now, let's write it in general form. We start from the point-slope form and manipulate it:
\[
y - 9 = \frac{5}{8}(x + 6)
\]
Multiply both sides by 8 to eliminate the fraction:
\[
8(y - 9) = 5(x + 6)
\]
Distributing both sides results in:
\[
8y - 72 = 5x + 30
\]
Rearranging this gives:
\[
-5x + 8y - 72 - 30 = 0
\]
or
\[
-5x + 8y - 102 = 0
\]
Multiplying through by -1 to make the leading coefficient positive gives the general form:
\[
5x - 8y + 102 = 0
\]
### Final Answers:
**Point-Slope Form:**
\[
y - 9 = \frac{5}{8}(x + 6)
\]
**General Form:**
\[
5x - 8y + 102 = 0
\]