Answer :
To find the equation of a line that is parallel to a given line and has an [tex]$x$[/tex]-intercept of 4, we need to understand a few key concepts about parallel lines and intercepts.
### Step-by-Step Solution:
1. Understanding Parallel Lines:
- Two lines are parallel if they have the same slope, denoted by 'm'. Therefore, the equation of any line parallel to the given line will have the form [tex]\(y = mx + c\)[/tex], where 'm' is the slope.
2. Identify the Slope of the Given Line:
- Since the equation of the line provided in the problem says "a line parallel to y = mx + c," we can infer that the slope 'm' of the given line is the same as the slope 'm' we need for our new line, as parallel lines share the same slope.
3. Determine the [tex]$x$[/tex]-intercept:
- The [tex]$x$[/tex]-intercept of a line is the point where the line intersects the [tex]$x$[/tex]-axis. This happens when [tex]\(y = 0\)[/tex]. For this line, the [tex]$x$[/tex]-intercept is given as 4. This means when [tex]\(y = 0\)[/tex], [tex]\(x = 4\)[/tex].
4. Form the Equation Using the [tex]$x$[/tex]-intercept:
- To use the [tex]$x$[/tex]-intercept in forming the equation of the line, we start with the general form [tex]\(y = mx + c\)[/tex] where 'm' is known and we need to find 'c'.
- At the [tex]$x$[/tex]-intercept, [tex]\(y = 0\)[/tex].
[tex]\[ 0 = m \cdot 4 + c \][/tex]
- Solving for [tex]\(c\)[/tex]:
[tex]\[ c = -4m \][/tex]
5. Write the Equation of the Parallel Line:
- Using the slope 'm' from the original line and the calculated intercept 'c', the equation of the line parallel to the given line with an [tex]$x$[/tex]-intercept of 4 will be:
[tex]\[ y = mx - 4m \][/tex]
So, the equation of the line parallel to the given line with an [tex]$x$[/tex]-intercept of 4 is:
[tex]\[ y = mx - 4m \][/tex]
Therefore, the final equation is:
[tex]\[ \boxed{y = mx - 4m} \][/tex]
This equation maintains the slope 'm' of the original line and adjusts the y-intercept based on the [tex]$x$[/tex]-intercept given.
### Step-by-Step Solution:
1. Understanding Parallel Lines:
- Two lines are parallel if they have the same slope, denoted by 'm'. Therefore, the equation of any line parallel to the given line will have the form [tex]\(y = mx + c\)[/tex], where 'm' is the slope.
2. Identify the Slope of the Given Line:
- Since the equation of the line provided in the problem says "a line parallel to y = mx + c," we can infer that the slope 'm' of the given line is the same as the slope 'm' we need for our new line, as parallel lines share the same slope.
3. Determine the [tex]$x$[/tex]-intercept:
- The [tex]$x$[/tex]-intercept of a line is the point where the line intersects the [tex]$x$[/tex]-axis. This happens when [tex]\(y = 0\)[/tex]. For this line, the [tex]$x$[/tex]-intercept is given as 4. This means when [tex]\(y = 0\)[/tex], [tex]\(x = 4\)[/tex].
4. Form the Equation Using the [tex]$x$[/tex]-intercept:
- To use the [tex]$x$[/tex]-intercept in forming the equation of the line, we start with the general form [tex]\(y = mx + c\)[/tex] where 'm' is known and we need to find 'c'.
- At the [tex]$x$[/tex]-intercept, [tex]\(y = 0\)[/tex].
[tex]\[ 0 = m \cdot 4 + c \][/tex]
- Solving for [tex]\(c\)[/tex]:
[tex]\[ c = -4m \][/tex]
5. Write the Equation of the Parallel Line:
- Using the slope 'm' from the original line and the calculated intercept 'c', the equation of the line parallel to the given line with an [tex]$x$[/tex]-intercept of 4 will be:
[tex]\[ y = mx - 4m \][/tex]
So, the equation of the line parallel to the given line with an [tex]$x$[/tex]-intercept of 4 is:
[tex]\[ y = mx - 4m \][/tex]
Therefore, the final equation is:
[tex]\[ \boxed{y = mx - 4m} \][/tex]
This equation maintains the slope 'm' of the original line and adjusts the y-intercept based on the [tex]$x$[/tex]-intercept given.
