What is the equation of the line that is parallel to the line $y-1=4(x+3)$ and passes through the point $(4,32)$?
A. $y=\frac{1}{4} x+33$
B. $y=-\frac{1}{4} x+36$
C. $y=4 x-16$
D. $y=4 x+16$
Answer (1)
Find the slope of the given line: y − 1 = 4 ( x + 3 ) becomes y = 4 x + 13 , so the slope is 4 .
Parallel lines have the same slope, so the new line also has a slope of 4 .
Use the point-slope form with the point ( 4 , 32 ) : y − 32 = 4 ( x − 4 ) .
Simplify to slope-intercept form: y = 4 x + 16 , so the final answer is y = 4 x + 16 .
Explanation
Understanding the Problem We are given the equation of a line y − 1 = 4 ( x + 3 ) and a point ( 4 , 32 ) . We want to find the equation of the line that is parallel to the given line and passes through the given point.
Finding the Slope First, let's rewrite the given equation in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept. We have
y − 1 = 4 ( x + 3 )
y − 1 = 4 x + 12
y = 4 x + 13
So, the slope of the given line is 4 .
Determining the Parallel Slope Since parallel lines have the same slope, the slope of the line we want to find is also 4 .
Applying Point-Slope Form Now we use the point-slope form of a line, y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is the given point ( 4 , 32 ) . Substituting the slope m = 4 and the point ( 4 , 32 ) , we get
y − 32 = 4 ( x − 4 )
Simplifying the Equation Simplify the equation to get the slope-intercept form:
y − 32 = 4 x − 16
y = 4 x − 16 + 32
y = 4 x + 16
Final Answer Therefore, the equation of the line that is parallel to the line y − 1 = 4 ( x + 3 ) and passes through the point ( 4 , 32 ) is y = 4 x + 16 .
Examples
Imagine you're designing a ramp for a skateboard park. You need the ramp to have the same steepness (slope) as another ramp but start at a different point. This problem helps you find the equation of the new ramp, ensuring it's parallel to the existing one and passes through the desired starting point. Understanding parallel lines is crucial in various real-world applications, from architecture and engineering to computer graphics and video game design.
