To find the equation of a line parallel to a given line, we need to ensure that both lines have the same slope. The general form 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.
Even though the specific slope [tex]\( m \)[/tex] and y-intercept [tex]\( b \)[/tex] of the original line are not explicitly given, we will proceed with the general strategy for writing the equation of a parallel line.
To find the equation of a line parallel to a given line and passing through a specific x-intercept, let's consider the key points:
1. Identifying the Slope:
Since the lines are parallel, the new line will have the same slope [tex]\( m \)[/tex] as the original line.
2. Using the X-Intercept:
An x-intercept of a line is the point where the line crosses the x-axis. At this point, the y-coordinate is zero. Let's denote the x-intercept as [tex]\( (4, 0) \)[/tex] (since the given x-intercept is 4). We can use this point to find the y-intercept [tex]\( b \)[/tex].
Steps to find the equation of the new line:
1. Write the general form of the equation using the known slope [tex]\( m \)[/tex]:
[tex]\[ y = mx + b \][/tex]
2. Substitute the coordinates of the x-intercept [tex]\( (4, 0) \)[/tex] into the equation to find [tex]\( b \)[/tex]:
[tex]\[ 0 = m(4) + b \][/tex]
[tex]\[ 0 = 4m + b \][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[ b = -4m \][/tex]
3. Substitute the value of [tex]\( b \)[/tex] back into the equation of the line:
[tex]\[ y = mx - 4m \][/tex]
Therefore, the equation of the line parallel to the given line with an x-intercept of 4 is:
[tex]\[ y = mx - 4m \][/tex]
Here, [tex]\( m \)[/tex] is the slope of the original line, which remains the same for the parallel line.
So, the final form of the equation is:
[tex]\[ y = m(x - 4) \][/tex]
This clearly shows the line is parallel to the original line with the given x-intercept.
