Let’s go through each step to carefully derive the equation of the parallel line passing through the point [tex]\((4, -1)\)[/tex].
### Step 1: Find the slope of the line [tex]\(2x - 3y = 9\)[/tex]
To find the slope, we first need to put the given equation in the form [tex]\(y = mx + b\)[/tex] (i.e., the slope-intercept form).
Starting with the equation:
[tex]\[2x - 3y = 9\][/tex]
We solve for [tex]\(y\)[/tex]:
[tex]\[ -3y = -2x + 9 \][/tex]
[tex]\[ y = \frac{2}{3}x - 3 \][/tex]
Thus, the slope [tex]\(m\)[/tex] of the line [tex]\(2x - 3y = 9\)[/tex] is:
[tex]\[ m = 2 / 3 \][/tex]
### Step 2: Use the slope to find the [tex]\(y\)[/tex]-intercept of the parallel line
Since parallel lines have the same slope, the slope [tex]\(m\)[/tex] of the parallel line is also:
[tex]\[ m = 2 / 3 \][/tex]
Now, we use the point [tex]\((4, -1)\)[/tex] to find the [tex]\(y\)[/tex]-intercept [tex]\(b\)[/tex] of the parallel line equation [tex]\(y = mx + b\)[/tex].
Plugging in the given point into the slope-intercept form, with [tex]\(x = 4\)[/tex], [tex]\(y = -1\)[/tex], and [tex]\(m = 2 / 3\)[/tex]:
[tex]\[ -1 = \frac{2}{3} \cdot 4 + b \][/tex]
First, calculate [tex]\(\frac{2}{3} \cdot 4\)[/tex]:
[tex]\[ \frac{2}{3} \cdot 4 = \frac{8}{3} \][/tex]
Then, substitute [tex]\(\frac{8}{3}\)[/tex] back into the equation:
[tex]\[ -1 = \frac{8}{3} + b \][/tex]
Next, solve for [tex]\(b\)[/tex]:
[tex]\[ b = -1 - \frac{8}{3} \][/tex]
[tex]\[ b = - \frac{3}{3} - \frac{8}{3} \][/tex]
[tex]\[b = - \frac{11}{3}\][/tex]
Thus, the [tex]\(y\)[/tex]-intercept [tex]\(b\)[/tex] of the parallel line is:
[tex]\[ b = -11 / 3 \][/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]
Using the slope [tex]\(m = 2 / 3\)[/tex] and the [tex]\(y\)[/tex]-intercept [tex]\(b = -11 / 3\)[/tex], the equation of the line is:
[tex]\[ y = \frac{2}{3}x - \frac{11}{3} \][/tex]
So the final equation is:
[tex]\[ y = \frac{2}{3}x - \frac{11}{3} \][/tex]
Putting it together in the format required:
### Solution
Step 1: Find the slope of the line [tex]\(2x - 3y = 9\)[/tex]
[tex]\[ m = 2 / 3 \][/tex]
Step 2: Use the slope to find the [tex]\(y\)[/tex]-intercept of the parallel line.
[tex]\[ b = -11 / 3 \][/tex]
Step 3: Write the equation of the line that passes through the point [tex]\((4, -1)\)[/tex] that is parallel to the line [tex]\(2x - 3y = 9\)[/tex]
[tex]\[ y = \frac{2}{3} x - \frac{11}{3} \][/tex]
