Questions: Use the given conditions to write an equation for the line passing through (-6,4) and parallel to the line whose equation is 5x-7y-6=0 The equation of the line is

Use the given conditions to write an equation for the line passing through (-6,4) and parallel to the line whose equation is 5x-7y-6=0

The equation of the line is

Transcript text: Use the given conditions to write an equation for the line Passing through $(-6,4)$ and parallel to the line whose equation is $5 x-7 y-6=0$ The equation of the line is $\square$

Solution

Solution Steps

To find the equation of a line passing through a given point and parallel to another line, we need to follow these steps:

Determine the slope of the given line.
Use the point-slope form of the equation of a line with the given point and the slope from step 1.
Simplify the equation to the desired form.

Step 1: Determine the Slope of the Given Line

The given line equation is $5x - 7y - 6 = 0$. The general form of a line is $Ax + By + C = 0$, and the slope of the line is $-\frac{A}{B}$.

For the given line:

$A = 5$
$B = -7$

Thus, the slope $m$ is: \[ m = -\frac{A}{B} = -\frac{5}{-7} = \frac{5}{7} \approx 0.7143 \]

Step 2: Use the Point-Slope Form

We need to find the equation of a line passing through the point $(-6, 4)$ with the slope $\frac{5}{7}$. The point-slope form of the equation of a line is: \[ y - y_1 = m(x - x_1) \]

Substituting the given point $(-6, 4)$ and the slope $\frac{5}{7}$: \[ y - 4 = \frac{5}{7}(x + 6) \]

Step 3: Simplify the Equation

Simplify the equation to the standard form $Ax + By + C = 0$: \[ y - 4 = \frac{5}{7}x + \frac{30}{7} \]

Multiply through by 7 to clear the fraction: \[ 7(y - 4) = 5x + 30 \] \[ 7y - 28 = 5x + 30 \]

Rearrange to the standard form: \[ -5x + 7y - 58 = 0 \]