Answer :
To find the equation of a line that is parallel to the given line \( y = 4x + 4 \) and has an \( x \)-intercept of 4, we will follow a series of logical steps:
### Step 1: Identify the Slope of the Given Line
First, note the slope of the given line. The equation \( y = 4x + 4 \) is in slope-intercept form \( y = mx + b \), where \( m \) is the slope. Here, the slope \( m \) is 4.
### Step 2: Understand the Properties of Parallel Lines
Parallel lines share the same slope. Therefore, the slope of our new line will also be 4.
### Step 3: Determine the Equation Format
We can start with the general form of a line equation in slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
### Step 4: Use the X-Intercept Information
Given that the \( x \)-intercept of our new line is 4, this means that when \( x = 4 \), \( y \) will be 0 (as this is the point of intercept with the x-axis).
### Step 5: Plug in the Point to Solve for y-intercept
We substitute \( x = 4 \) and \( y = 0 \) into the equation \( y = mx + b \):
[tex]\[ 0 = 4(4) + b \][/tex]
[tex]\[ 0 = 16 + b \][/tex]
### Step 6: Solve for \( b \)
To find \( b \), solve the above equation:
[tex]\[ b = -16 \][/tex]
### Step 7: Write the Equation of the Parallel Line
Now that we have the slope \( m = 4 \) and the y-intercept \( b = -16 \), we can write the equation of our new parallel line as:
[tex]\[ y = 4x - 16 \][/tex]
Therefore, the equation of the line parallel to the given line \( y = 4x + 4 \) with an \( x \)-intercept of 4 is:
[tex]\[ \boxed{y = 4x - 16} \][/tex]
### Step 1: Identify the Slope of the Given Line
First, note the slope of the given line. The equation \( y = 4x + 4 \) is in slope-intercept form \( y = mx + b \), where \( m \) is the slope. Here, the slope \( m \) is 4.
### Step 2: Understand the Properties of Parallel Lines
Parallel lines share the same slope. Therefore, the slope of our new line will also be 4.
### Step 3: Determine the Equation Format
We can start with the general form of a line equation in slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
### Step 4: Use the X-Intercept Information
Given that the \( x \)-intercept of our new line is 4, this means that when \( x = 4 \), \( y \) will be 0 (as this is the point of intercept with the x-axis).
### Step 5: Plug in the Point to Solve for y-intercept
We substitute \( x = 4 \) and \( y = 0 \) into the equation \( y = mx + b \):
[tex]\[ 0 = 4(4) + b \][/tex]
[tex]\[ 0 = 16 + b \][/tex]
### Step 6: Solve for \( b \)
To find \( b \), solve the above equation:
[tex]\[ b = -16 \][/tex]
### Step 7: Write the Equation of the Parallel Line
Now that we have the slope \( m = 4 \) and the y-intercept \( b = -16 \), we can write the equation of our new parallel line as:
[tex]\[ y = 4x - 16 \][/tex]
Therefore, the equation of the line parallel to the given line \( y = 4x + 4 \) with an \( x \)-intercept of 4 is:
[tex]\[ \boxed{y = 4x - 16} \][/tex]
