To determine the equation of a line parallel to a given line, we should start by understanding that parallel lines have the same slope. Suppose the given line has the equation [tex]\( y = mx + c \)[/tex].
Given information:
1. The parallel line must have the same slope [tex]\( m \)[/tex]. Let's denote the slope as [tex]\( m = 2 \)[/tex].
2. The parallel line has an [tex]\(x\)[/tex]-intercept of 4, which means the line passes through the point (4, 0).
We need to find the [tex]\( y \)[/tex]-intercept, [tex]\( b \)[/tex], of the new line that has this [tex]\( x \)[/tex]-intercept and the same slope as the given line.
Step-by-step solution:
1. Start with the slope-intercept form of the line: [tex]\( y = mx + b \)[/tex].
2. Use the slope [tex]\( m = 2 \)[/tex] (which is the same as the slope of the original line) in the equation:
[tex]\[ y = 2x + b \][/tex]
3. Since the line passes through the point (4, 0) (given as the [tex]\( x \)[/tex]-intercept):
[tex]\[ 0 = 2(4) + b \][/tex]
4. Solve for [tex]\( b \)[/tex]:
[tex]\[ 0 = 8 + b \][/tex]
[tex]\[ b = -8 \][/tex]
So, the equation of the line parallel to the given line and with an [tex]\( x \)[/tex]-intercept of 4 is:
[tex]\[ y = 2x - 8 \][/tex]
