Sure, let's find the equation of a line that is parallel to [tex]\(y = -3x + 7\)[/tex] and passes through the point [tex]\((2, -4)\)[/tex].
1. Identify the slope of the original line:
The original line is given in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope. In [tex]\(y = -3x + 7\)[/tex], the slope [tex]\(m\)[/tex] is [tex]\(-3\)[/tex].
2. Understand that parallel lines have the same slope:
Since parallel lines share the same slope, the slope of our new line will also be [tex]\(-3\)[/tex].
3. Use the point-slope form of a line equation:
We know the slope [tex]\(m = -3\)[/tex] and a point [tex]\( (2, -4) \)[/tex] through which the line passes. The point-slope form of the equation is
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is the given point. Plugging in the values, we get:
[tex]\[
y - (-4) = -3(x - 2)
\][/tex]
4. Simplify the equation:
[tex]\[
y + 4 = -3(x - 2)
\][/tex]
Distribute the [tex]\(-3\)[/tex]:
[tex]\[
y + 4 = -3x + 6
\][/tex]
Subtract 4 from both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = -3x + 6 - 4
\][/tex]
[tex]\[
y = -3x + 2
\][/tex]
So, the equation of the line parallel to [tex]\(y = -3x + 7\)[/tex] that passes through [tex]\((2, -4)\)[/tex] in slope-intercept form is:
[tex]\[
y = -3x + 2
\][/tex]
Among the options provided, the correct answer is:
C. [tex]\(y = -3x + 2\)[/tex]
