Sure, let's determine the equation of the line that passes through the point [tex]\((3, 7)\)[/tex] and is parallel to the line [tex]\(y = -3x + 6\)[/tex].
1. Identify the slope:
Because the two lines are parallel, they must have the same slope. From the given line [tex]\(y = -3x + 6\)[/tex], we see that the slope is [tex]\(-3\)[/tex].
2. Form of the equation:
The equation of the required line will be in the slope-intercept form, [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
3. Plug in the slope:
Since the lines are parallel, the slope [tex]\(m\)[/tex] will be [tex]\(-3\)[/tex]. So the equation of our line is:
[tex]\[
y = -3x + b
\][/tex]
4. Use the given point to find [tex]\(b\)[/tex]:
We will use the point [tex]\((3, 7)\)[/tex] that the line passes through to find the y-intercept [tex]\(b\)[/tex].
Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = 7\)[/tex] into the equation:
[tex]\[
7 = -3(3) + b
\][/tex]
Simplify:
[tex]\[
7 = -9 + b
\][/tex]
5. Solve for [tex]\(b\)[/tex]:
Add 9 to both sides of the equation:
[tex]\[
7 + 9 = b
\][/tex]
[tex]\[
b = 16
\][/tex]
6. Write the final equation:
So, substituting the values of [tex]\(m\)[/tex] and [tex]\(b\)[/tex] back into the slope-intercept form, the equation of the line is:
[tex]\[
y = -3x + 16
\][/tex]
The line that passes through the point [tex]\((3, 7)\)[/tex] and is parallel to [tex]\(y = -3x + 6\)[/tex] has the equation:
[tex]\[
y = -3x + 16
\][/tex]
