Questions: What is an equation of the line that passes through the point (3,2) and is parallel to the line 2x+3y=24?

Find the equation of the line that passes through the point \((3, 2)\) and is parallel to the line \(2x + 3y = 24\).

Find the slope of the given line.

Rewrite the given line \(2x + 3y = 24\) in slope-intercept form \(y = mx + b\): \[ 3y = -2x + 24 \\ y = -\frac{2}{3}x + 8 \] The slope \(m\) of the given line is \(-\frac{2}{3}\).

Determine the slope of the parallel line.

Slope of parallel lines.

Parallel lines have the same slope. Therefore, the slope of the desired line is also \(-\frac{2}{3}\).

Write the equation of the desired line using the point-slope form.

Use the point-slope form.

The point-slope form is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (3, 2)\) and \(m = -\frac{2}{3}\): \[ y - 2 = -\frac{2}{3}(x - 3) \] Simplify the equation: \[ y - 2 = -\frac{2}{3}x + 2 \\ y = -\frac{2}{3}x + 4 \]

Convert the equation to standard form.

Convert to standard form \(Ax + By = C\).

Multiply all terms by 3 to eliminate the fraction: \[ 3y = -2x + 12 \\ 2x + 3y = 12 \]

The equation of the line is \(\boxed{2x + 3y = 12}\).