To find the equation of the line parallel to a given line and with a specific [tex]\( x \)[/tex]-intercept, we can follow these steps:
1. Identify the Slope of the Given Line:
Since the new line is parallel to the given line, it must have the same slope [tex]\( m \)[/tex]. Let's assume the slope [tex]\( m \)[/tex] of the given line is 1 (as no specific line is given).
2. Determine the [tex]\( x \)[/tex]-intercept:
The problem states that the [tex]\( x \)[/tex]-intercept of the new line is 4. This means the line passes through the point [tex]\( (4, 0) \)[/tex].
3. Use the Point-Slope Form to Find the y-intercept:
Recall the point-slope form of a linear equation:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, we can use the point [tex]\( (4, 0) \)[/tex] and the slope [tex]\( m = 1 \)[/tex]:
[tex]\[
y - 0 = 1(x - 4)
\][/tex]
Simplifying this, we get:
[tex]\[
y = 1(x - 4)
\][/tex]
[tex]\[
y = x - 4
\][/tex]
4. Identify the y-intercept:
From the equation [tex]\( y = x - 4 \)[/tex], we observe that the constant term [tex]\(-4\)[/tex] is the [tex]\( y \)[/tex]-intercept.
5. Write the final equation:
Therefore, the equation of the line parallel to the given line and with an [tex]\( x \)[/tex]-intercept of 4 is:
[tex]\[
y = 1x - 4
\][/tex]
So, the equation of the line is:
[tex]\[
y = x - 4
\][/tex]
