To find the equation of a line parallel to a given line with an x-intercept of 4, follow these steps:
1. Identify the Slope ([tex]\( m \)[/tex]) of the Given Line:
- Let the equation of the given line be in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Since we need a parallel line, the slope ([tex]\( m \)[/tex]) of our new line must be the same as the given line.
2. Find the X-Intercept and Determine the Y-Intercept ([tex]\( c \)[/tex]) of the New Line:
- By definition, the x-intercept is the point where the line crosses the x-axis ([tex]\( y = 0 \)[/tex]), so for the x-intercept of 4, the point is [tex]\( (4, 0) \)[/tex].
- Substitute [tex]\( x = 4 \)[/tex] and [tex]\( y = 0 \)[/tex] into the line equation [tex]\( y = mx + c \)[/tex]:
[tex]\[
0 = m \cdot 4 + c
\][/tex]
- Solve for [tex]\( c \)[/tex]:
[tex]\[
c = -4m
\][/tex]
3. Write the Equation of the New Line:
- Now that we have both the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( c = -4m \)[/tex], we can write the equation of the new parallel 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]
Thus, filling in the blanks:
[tex]\[
y = \boxed{m} x + \boxed{-4m}
\][/tex]
