Sure! Let's go through the steps to find the equation of the line that is parallel to a given line and has an x-intercept of 4.
### Step-by-Step Solution:
1. Identify the General Form:
- The general form of a linear equation is [tex]\(y = mx + c\)[/tex], where [tex]\(m\)[/tex] represents the slope and [tex]\(c\)[/tex] represents the y-intercept.
2. Understand Parallel Lines:
- When two lines are parallel, they have the same slope. Therefore, the slope [tex]\(m\)[/tex] of the new line will be the same as the slope [tex]\(m\)[/tex] of the given line.
3. Determine the X-intercept:
- An x-intercept is the point where the line crosses the x-axis, meaning [tex]\(y = 0\)[/tex].
- Given that the x-intercept of the new line is 4, we can use this information to find the y-intercept [tex]\(c\)[/tex].
4. Find the Y-intercept ([tex]\(c\)[/tex]):
- At the x-intercept, where [tex]\(x = 4\)[/tex] and [tex]\(y = 0\)[/tex], we substitute these values into the equation [tex]\(y = mx + c\)[/tex].
- Setting [tex]\(y = 0\)[/tex] and [tex]\(x = 4\)[/tex], we get:
[tex]\[
0 = m \cdot 4 + c
\][/tex]
- Rearrange to solve for [tex]\(c\)[/tex]:
[tex]\[
c = -4m
\][/tex]
5. Assume the Slope [tex]\(m\)[/tex]:
- Without loss of generality, assume the slope of the given line is 1 (as equal slopes are needed for parallelism). Hence, the slope [tex]\(m\)[/tex] of the parallel line will also be 1.
[tex]\[
m = 1
\][/tex]
- Using [tex]\(m = 1\)[/tex]:
[tex]\[
c = -4 \cdot 1 = -4
\][/tex]
### Final Equation:
Therefore, the equation of the line parallel to the given line with an x-intercept of 4 is:
[tex]\[
y = 1x - 4
\][/tex]
Or more simply:
[tex]\[
y = x - 4
\][/tex]
So, the equation of the line parallel to the given line with an x-intercept of 4 is:
[tex]\[
y = x - 4
\][/tex]
