Questions: Find an equation of the line. Write the equation in the standard form. Through (5,8); parallel to 3x-y=6. (Use integers or fractions for any numbers in the equation. Simplify your answer.)

$Find an equation of the line. Write the equation in the standard form. Through (5,8); parallel to 3x-y=6. (Use integers or fractions for any numbers in the equation. Simplify your answer.)$

Transcript text: Find an equation of the line. Write the equation in the standard form. Through $(5,8)$; parallel to $3 x-y=6$. (Use integers or fractions for any numbers in the equation. Simplify your answer.)

Solution

Solution Steps

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

Identify the slope of the given line.
Use the point-slope form of the equation of a line with the identified slope and the given point.
Convert the equation to standard form.

Step 1: Identify the Slope

The given line is represented by the equation $3x - y = 6$. To find the slope, we can rewrite this in slope-intercept form $y = mx + b$: \[ y = 3x - 6 \] From this, we see that the slope $m$ of the line is $3$.

Step 2: Use Point-Slope Form

We need to find the equation of a line that is parallel to the given line and passes through the point $(5, 8)$. Using the point-slope form of the equation: \[ y - y_1 = m(x - x_1) \] Substituting $m = 3$, $x_1 = 5$, and $y_1 = 8$: \[ y - 8 = 3(x - 5) \]

Step 3: Convert to Standard Form

Expanding the equation: \[ y - 8 = 3x - 15 \] Rearranging gives: \[ 3x - y = 7 \] This is now in standard form $Ax + By = C$.