To find the equation of a line that contains the point [tex]\((2, -5)\)[/tex] and is parallel to the line [tex]\(y = 3x - 4\)[/tex], follow these steps:
1. Identify the slope of the parallel line:
Since the line is parallel to [tex]\(y = 3x - 4\)[/tex], it will have the same slope. The slope of this line is [tex]\(3\)[/tex].
2. Formulate the equation of the new line:
The general equation of a line in slope-intercept form is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
3. Use the given point to find the y-intercept ([tex]\(b\)[/tex]):
Plug in the coordinates of the given point [tex]\((2, -5)\)[/tex] into the slope-intercept form, [tex]\(y = mx + b\)[/tex].
[tex]\[
-5 = 3 \cdot 2 + b
\][/tex]
4. Solve for [tex]\(b\)[/tex]:
[tex]\[
-5 = 6 + b
\][/tex]
[tex]\[
b = -5 - 6
\][/tex]
[tex]\[
b = -11
\][/tex]
5. Write the final equation:
Now we have the slope [tex]\(m = 3\)[/tex] and the y-intercept [tex]\(b = -11\)[/tex]. Therefore, the equation of the line is:
[tex]\[
y = 3x - 11
\][/tex]
Therefore, the correct equation of the line that contains the point [tex]\((2, -5)\)[/tex] and is parallel to the line [tex]\(y = 3x - 4\)[/tex] is:
[tex]\[ y = 3x - 11 \][/tex]
The answer is:
C. [tex]\(y = 3x - 11\)[/tex]
