Select the correct answer.
What is the equation of the line that passes through the point $(6,11)$ and is parallel to the line with the following equation?
$y=\frac{2}{3} x-5$
A. $y=-\frac{3}{2} x+17$
B. $y=-\frac{3}{2} x+5$
C. $y=\frac{2}{3} x+7$
D. $y=\frac{2}{3} x-15$
Answer (1)
The line is parallel to y = 3 2 x − 5 , so it has the same slope: m = 3 2 .
The equation of the line is of the form y = 3 2 x + b .
Substitute the point ( 6 , 11 ) into the equation to solve for b : 11 = 3 2 ( 6 ) + b .
Simplify the equation to find the value of b : b = 7 . The equation of the line is y = 3 2 x + 7 .
Explanation
Understanding the Problem We are given a point ( 6 , 11 ) and a line y = 3 2 x − 5 . We need to find the equation of the line that passes through the given point and is parallel to the given line.
Finding the Slope Since the line we are looking for is parallel to the line y = 3 2 x − 5 , it has the same slope. The slope of the given line is 3 2 . Therefore, the slope of the line we are looking for is also 3 2 .
Using the Point-Slope Form The equation of the line we are looking for is of the form y = 3 2 x + b , where b is the y-intercept. To find the value of b , we can substitute the given point ( 6 , 11 ) into the equation: 11 = 3 2 ( 6 ) + b
Solving for the y-intercept Now, we solve for b : 11 = 3 2 ( 6 ) + b 11 = 4 + b b = 11 − 4 b = 7
Writing the Equation of the Line Therefore, the equation of the line is y = 3 2 x + 7 .
Final Answer The correct answer is C. y = 3 2 x + 7 .
Examples
Imagine you're designing a ramp for a skateboard park. You want the ramp to have the same steepness (slope) as another ramp but start at a different height. This problem helps you find the equation of the new ramp, ensuring it's parallel to the existing one and passes through a specific point (the starting height). Understanding parallel lines and their equations is crucial in various fields like architecture, engineering, and even computer graphics, where maintaining consistent angles and slopes is essential for creating accurate designs and simulations.
