To find the equation of a line that is parallel to [tex]\(-2x + 3y = -6\)[/tex] and passes through the point [tex]\((-2, 0)\)[/tex], follow these steps:
1. Determine the slope of the given line:
The given equation of the line is [tex]\(-2x + 3y = -6\)[/tex].
To find the slope, we first convert this equation to the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
[tex]\[
-2x + 3y = -6
\][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
3y = 2x - 6
\][/tex]
[tex]\[
y = \frac{2}{3}x - 2
\][/tex]
So, the slope [tex]\( m \)[/tex] of the given line is [tex]\(\frac{2}{3}\)[/tex].
2. Construct the equation of the new line:
Since parallel lines have the same slope, the slope of the new line is also [tex]\(\frac{2}{3}\)[/tex].
Use the point-slope form of the equation of a line, [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( (x_1, y_1) \)[/tex] is the point the line passes through.
Given point [tex]\((-2, 0)\)[/tex]:
[tex]\[
y - 0 = \frac{2}{3}(x - (-2))
\][/tex]
Simplify the equation:
[tex]\[
y = \frac{2}{3}(x + 2)
\][/tex]
3. Convert the equation to the standard form:
Expand and simplify the above equation:
[tex]\[
y = \frac{2}{3}x + \frac{4}{3}
\][/tex]
To eliminate the fraction, multiply through by 3:
[tex]\[
3y = 2x + 4
\][/tex]
Rearrange to the standard form [tex]\( Ax + By = C \)[/tex]:
[tex]\[
2x - 3y = -4
\][/tex]
Thus, the equation of the line that is parallel to [tex]\(-2x + 3y = -6\)[/tex] and passes through the point [tex]\((-2, 0)\)[/tex] is:
[tex]\[
2x - 3y = -4
\][/tex]
