Let's work through the problem step by step.
Problem:
Find the equation of the line passing through the point [tex]\((4, -1)\)[/tex] that is parallel to the line [tex]\(2x - 3y = 9\)[/tex].
Solution:
Step 1: Find the slope of the line [tex]\(2x - 3y = 9\)[/tex].
The general form of a line is [tex]\(Ax + By = C\)[/tex]. We can rewrite it as [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
For the line [tex]\(2x - 3y = 9\)[/tex]:
- [tex]\(A = 2\)[/tex]
- [tex]\(B = -3\)[/tex]
The slope [tex]\(m\)[/tex] is given by [tex]\(m = -A / B\)[/tex].
So,
[tex]\[ m = -\frac{2}{-3} = \frac{2}{3} \][/tex]
Since parallel lines have the same slope, the slope of the line parallel to [tex]\(2x - 3y = 9\)[/tex] is also:
[tex]\[ m = \frac{2}{3} \][/tex]
Step 2: Use the slope to find the [tex]\(y\)[/tex]-intercept of the parallel line.
We need to use the point [tex]\( (4, -1) \)[/tex] and the slope [tex]\( \frac{2}{3} \)[/tex] to find the [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex]. The equation of the line is in the form [tex]\( y = mx + b \)[/tex].
Substitute the point and the slope into [tex]\( y = mx + b \)[/tex]:
[tex]\[ -1 = \left(\frac{2}{3}\right) \cdot 4 + b \][/tex]
Simplify and solve for [tex]\( b \)[/tex]:
[tex]\[ -1 = \frac{8}{3} + b \][/tex]
[tex]\[ b = -1 - \frac{8}{3} \][/tex]
[tex]\[ b = -\frac{3}{3} - \frac{8}{3} \][/tex]
[tex]\[ b = -\frac{11}{3} \][/tex]
So the [tex]\( y \)[/tex]-intercept is:
[tex]\[ b = -\frac{11}{3} \][/tex]
Step 3: Write the equation of the line that passes through the point [tex]\((4, -1)\)[/tex] and is parallel to the line [tex]\(2x - 3y = 9\)[/tex].
The equation of the parallel line with slope [tex]\( \frac{2}{3} \)[/tex] and [tex]\( y \)[/tex]-intercept [tex]\( -\frac{11}{3} \)[/tex] is in the form:
[tex]\[ y = mx + b \][/tex]
So, substitute [tex]\( m = \frac{2}{3} \)[/tex] and [tex]\( b = -\frac{11}{3} \)[/tex]:
[tex]\[ y = \frac{2}{3}x - \frac{11}{3} \][/tex]
Therefore, the equation of the line is:
[tex]\[ y = \frac{2}{3} x - \frac{11}{3} \][/tex]
So the detailed solution is:
Step 1:
The slope [tex]\( m = \frac{2}{3} \)[/tex]
Step 2:
The [tex]\( y \)[/tex]-intercept [tex]\( b = -\frac{11}{3} \)[/tex]
Step 3:
The equation of the line is:
[tex]\[ y = \frac{2}{3} x - \frac{11}{3} \][/tex]
