Answer :
Sure, let's find the equation of the line that is parallel to the given line [tex]\( y = -4x + 4 \)[/tex] and has an [tex]\( x \)[/tex]-intercept of 4.
### Step-by-Step Solution:
1. Identify the slope of the given line:
The equation of the given line is [tex]\( y = -4x + 4 \)[/tex]. This is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. From the equation, the slope [tex]\( m \)[/tex] is [tex]\(-4\)[/tex].
2. Determine the slope of the parallel line:
Parallel lines have identical slopes. Therefore, the slope of the new line is also [tex]\(-4\)[/tex].
3. Use the given [tex]\( x \)[/tex]-intercept to find the new line’s y-intercept:
The [tex]\( x \)[/tex]-intercept is the point where the line crosses the [tex]\( x \)[/tex]-axis, which means [tex]\( y = 0 \)[/tex] at this point. Given the [tex]\( x \)[/tex]-intercept is 4, the point on the line is [tex]\( (4, 0) \)[/tex].
4. Substitute the point and the slope into the slope-intercept equation:
The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex]. We know [tex]\( m = -4 \)[/tex] and we have the point [tex]\( (4, 0) \)[/tex].
Substitute [tex]\( x = 4 \)[/tex], [tex]\( y = 0 \)[/tex], and [tex]\( m = -4 \)[/tex] into the slope-intercept equation to find [tex]\( b \)[/tex]:
[tex]\[ y = mx + b \\ 0 = -4(4) + b \][/tex]
5. Solve for [tex]\( b \)[/tex]:
[tex]\[ 0 = -16 + b \\ b = 16 \][/tex]
6. Write the equation of the new line:
Substitute the slope [tex]\( m = -4 \)[/tex] and the y-intercept [tex]\( b = 16 \)[/tex] back into the slope-intercept form:
[tex]\[ y = -4x + 16 \][/tex]
Therefore, the equation of the line that is parallel to the line [tex]\( y = -4x + 4 \)[/tex] and has an [tex]\( x \)[/tex]-intercept of 4 is:
[tex]\[ y = -4x + 16 \][/tex]
### Step-by-Step Solution:
1. Identify the slope of the given line:
The equation of the given line is [tex]\( y = -4x + 4 \)[/tex]. This is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. From the equation, the slope [tex]\( m \)[/tex] is [tex]\(-4\)[/tex].
2. Determine the slope of the parallel line:
Parallel lines have identical slopes. Therefore, the slope of the new line is also [tex]\(-4\)[/tex].
3. Use the given [tex]\( x \)[/tex]-intercept to find the new line’s y-intercept:
The [tex]\( x \)[/tex]-intercept is the point where the line crosses the [tex]\( x \)[/tex]-axis, which means [tex]\( y = 0 \)[/tex] at this point. Given the [tex]\( x \)[/tex]-intercept is 4, the point on the line is [tex]\( (4, 0) \)[/tex].
4. Substitute the point and the slope into the slope-intercept equation:
The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex]. We know [tex]\( m = -4 \)[/tex] and we have the point [tex]\( (4, 0) \)[/tex].
Substitute [tex]\( x = 4 \)[/tex], [tex]\( y = 0 \)[/tex], and [tex]\( m = -4 \)[/tex] into the slope-intercept equation to find [tex]\( b \)[/tex]:
[tex]\[ y = mx + b \\ 0 = -4(4) + b \][/tex]
5. Solve for [tex]\( b \)[/tex]:
[tex]\[ 0 = -16 + b \\ b = 16 \][/tex]
6. Write the equation of the new line:
Substitute the slope [tex]\( m = -4 \)[/tex] and the y-intercept [tex]\( b = 16 \)[/tex] back into the slope-intercept form:
[tex]\[ y = -4x + 16 \][/tex]
Therefore, the equation of the line that is parallel to the line [tex]\( y = -4x + 4 \)[/tex] and has an [tex]\( x \)[/tex]-intercept of 4 is:
[tex]\[ y = -4x + 16 \][/tex]
