Answer
$y = -\frac{2}{3}x + \frac{10}{3}$
Work Step by Step
We know that two parallel lines have the same slope, so if we are given the equation of a line parallel to the one we are looking for, then we know we already have the slope of our unknown line.
The known line is written in standard form, which is given by the formula:
$Ax + Bx = C$
We need to rewrite the equation in slope-intercept form so we can easily locate the slope of the line.
So for $2x + 3y = 9$, we first subtract $2x$ from each side to isolate the $y$ term:
$3y = -2x + 9$
Divide all terms by $3$ to isolate $y$:
$y = -\frac{2}{3} + \frac{9}{3}$
Simplify the fraction by dividing the numerator by the denominator:
$y = -\frac{2}{3} + 3$
The slope is $-\frac{2}{3}$. This will also be the slope for our unknown line.
We are given the point $(-1, 4)$ and our slope $m = -\frac{2}{3}$. We now have the slope and a point on our unknown line.
We can plug these values into the point-slope equation, which is given by the formula:
$y - y_1 = m(x - x_1)$
Let's plug in the points and slope into the formula:
$y - 4 = -\frac{2}{3}(x - (-1))$
Use distribution to simplify:
$y - 4 = -\frac{2}{3}x - (-\frac{2}{3})(-1)$
Multiply to simplify:
$y - 4 = -\frac{2}{3}x - \frac{2}{3}$
To change this equation into point-intercept form, we need to isolate $y$. To isolate $y$, we add $4$ to each side of the equation:
$y = -\frac{2}{3}x - \frac{2}{3} + 4$
Change $4$ into an equivalent fraction that has $3$ as its denominator so that both fractions have the same denominator:
$y = -\frac{2}{3}x - \frac{2}{3} + \frac{12}{3}$
Add the fractions to simplify:
$y = -\frac{2}{3}x + \frac{10}{3}$
Now, we have the equation of the line in slope-intercept form.
