Answer :
To determine the equation of a line parallel to a given line that has an \( x \)-intercept of 4, we need to follow these steps:
1. Equation of the original line:
The original line is provided in the standard form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
2. Parallel line characteristics:
Parallel lines have the same slope (\( m \)). Therefore, the new line will have the same slope as the original line.
3. Determining the new line's intercept:
For our new line with an \( x \)-intercept of 4, by definition, the \( x \)-intercept is the point where the line crosses the x-axis (\( y = 0 \)). Hence, if we substitute \( x = 4 \) into the line's equation, we should get \( y = 0 \).
4. Finding the y-intercept (b):
Let's generalize the equation of the new line as \( y = mx + b \). Since we know it intercepts the x-axis at \( x = 4 \):
[tex]\[ 0 = m \cdot 4 + b \][/tex]
Solving for \( b \):
[tex]\[ 0 = 4m + b \implies b = -4m \][/tex]
Now, substituting \( b = -4m \) back into the line equation \( y = mx + b \), we get:
[tex]\[ y = mx - 4m \][/tex]
However, when we rewrite the final equation, the standard way to express a line equation is simplified based on the given information. If our aim is to find \( x \)- and \( y \)-coordinates where \( y = 0 \):
The final simplified form of any line parallel to the original line with an \( x \)-intercept of 4 will point to \( y = 0 \).
The resulting equation based on these steps is:
[tex]\[ y = 0 \][/tex]
1. Equation of the original line:
The original line is provided in the standard form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
2. Parallel line characteristics:
Parallel lines have the same slope (\( m \)). Therefore, the new line will have the same slope as the original line.
3. Determining the new line's intercept:
For our new line with an \( x \)-intercept of 4, by definition, the \( x \)-intercept is the point where the line crosses the x-axis (\( y = 0 \)). Hence, if we substitute \( x = 4 \) into the line's equation, we should get \( y = 0 \).
4. Finding the y-intercept (b):
Let's generalize the equation of the new line as \( y = mx + b \). Since we know it intercepts the x-axis at \( x = 4 \):
[tex]\[ 0 = m \cdot 4 + b \][/tex]
Solving for \( b \):
[tex]\[ 0 = 4m + b \implies b = -4m \][/tex]
Now, substituting \( b = -4m \) back into the line equation \( y = mx + b \), we get:
[tex]\[ y = mx - 4m \][/tex]
However, when we rewrite the final equation, the standard way to express a line equation is simplified based on the given information. If our aim is to find \( x \)- and \( y \)-coordinates where \( y = 0 \):
The final simplified form of any line parallel to the original line with an \( x \)-intercept of 4 will point to \( y = 0 \).
The resulting equation based on these steps is:
[tex]\[ y = 0 \][/tex]
