To determine the equation of Line A, we need to follow several steps using the basic principles of algebra.
1. Identify the Slope of the Given Line:
The given line equation is [tex]\( 3 - y = 3x \)[/tex]. To find the slope, we need to rewrite this equation in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope.
[tex]\[
3 - y = 3x
\][/tex]
Subtract 3 from both sides:
[tex]\[
-y = 3x - 3
\][/tex]
Multiply both sides by -1 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = -3x + 3
\][/tex]
So, the slope ([tex]\( m \)[/tex]) of the given line is -3.
2. Determine the Slope of Line A:
Since Line A is parallel to the given line, it will have the same slope. Therefore, the slope of Line A is also -3.
3. Use the Point-Slope Form to Find the Equation of Line A:
Line A passes through the point [tex]\((-6, 22)\)[/tex]. We use the point-slope form of the equation of a line:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] is the point [tex]\((-6, 22)\)[/tex] and [tex]\( m \)[/tex] is the slope -3.
Substituting the given values:
[tex]\[
y - 22 = -3(x + 6)
\][/tex]
4. Simplify the Equation:
Distribute the -3 on the right-hand side:
[tex]\[
y - 22 = -3x - 18
\][/tex]
Add 22 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = -3x + 4
\][/tex]
Therefore, the equation of Line A in the form [tex]\( y = mx + C \)[/tex] is:
[tex]\[
y = -3x + 4
\][/tex]
