What is the equation of a line that is parallel to [tex]\(-2x + 3y = -6\)[/tex] and passes through the point [tex]\((-2, 0)\)[/tex]?

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Answer :

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]