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. $y=4 x+16$

Find the slope of the given line by rewriting it in slope-intercept form: y = 4 x + 13 , so the slope is 4 .
Use the fact that parallel lines have the same slope, so the slope of the line we seek is also 4 .
Use the point-slope form of a line, y − y 1 ​ = m ( x − x 1 ​ ) , with the point ( 4 , 32 ) and slope 4 : y − 32 = 4 ( x − 4 ) .
Rewrite the equation in slope-intercept form: y = 4 x + 16 . 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 need 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 ) to find its slope. y − 1 = 4 ( x + 3 )
y − 1 = 4 x + 12
y = 4 x + 13 The slope of the given line is 4 .

Parallel Lines Since parallel lines have the same slope, the slope of the line we seek is also 4 .

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 ) into the point-slope form, we get: y − 32 = 4 ( x − 4 )

Slope-Intercept Form Next, we rewrite the equation in slope-intercept form, y = m x + b :
y − 32 = 4 ( x − 4 )
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 be parallel to a certain existing slope for a smooth transition. Knowing the slope of the existing feature and a point where you want your ramp to start, you can use the equation of a line parallel to another line to determine the exact path your ramp should follow. This ensures a consistent angle and a safe, fun ride!

Answered by GinnyAnswer | 2025-07-03

The equation of the line parallel to y − 1 = 4 ( x + 3 ) and passing through the point ( 4 , 32 ) is y = 4 x + 16 , which corresponds to option D. This is found by recognizing that parallel lines have the same slope and using the point-slope form of a line with given slope and point. Thus, the final answer is y = 4 x + 16 .
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Answered by Anonymous | 2025-07-04