Which equation represents a line that is parallel to the line that passes through the points $(-6,9)$ and $(7,-17)$?
A. $y=-\frac{1}{2} x+13$
B. $y=\frac{1}{2} x+13$
C. $y=2 x+13$
D. $y=-2 x+13
Answer (2)
Calculate the slope of the line passing through points ( − 6 , 9 ) and ( 7 , − 17 ) using the formula m = x 2 − x 1 y 2 − y 1 , which gives m = − 2 .
Recognize that parallel lines have the same slope, so the parallel line also has a slope of − 2 .
Compare the slopes of the given options to the calculated slope of − 2 .
Identify the equation with a slope of − 2 , which is y = − 2 x + 13 , making option D the correct answer. y = − 2 x + 13
Explanation
Calculate the slope First, we need to find the slope of the line that passes through the points ( − 6 , 9 ) and ( 7 , − 17 ) . The slope, m , is calculated using the formula: m = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of the two points.
Substitute the values Substituting the given points into the formula, we get: m = 7 − ( − 6 ) − 17 − 9 = 13 − 26 = − 2 So, the slope of the line passing through the points ( − 6 , 9 ) and ( 7 , − 17 ) is − 2 .
Find the slope of the parallel line Now, we need to find the equation of a line that is parallel to this line. Parallel lines have the same slope. Therefore, the slope of the parallel line must also be − 2 .
Analyze the options We are given four options: A. y = − 2 1 x + 13 B. y = 2 1 x + 13 C. y = 2 x + 13 D. y = − 2 x + 13
The equation of a line is in the slope-intercept form y = m x + b , where m is the slope and b is the y-intercept. We need to find the option where the slope m is equal to − 2 .
Compare the slopes and find the correct option Comparing the slopes of the given options with the slope of the parallel line (which is − 2 ): A. The slope is − 2 1 B. The slope is 2 1 C. The slope is 2 D. The slope is − 2
Option D has a slope of − 2 , which is the same as the slope of the line passing through the points ( − 6 , 9 ) and ( 7 , − 17 ) . Therefore, the equation that represents a line parallel to the line passing through the points ( − 6 , 9 ) and ( 7 , − 17 ) is y = − 2 x + 13 .
Final Answer Therefore, the equation of the line that is parallel to the line passing through the points ( − 6 , 9 ) and ( 7 , − 17 ) is: y = − 2 x + 13 So, the correct answer is D.
Examples
Understanding parallel lines is crucial in architecture and design. For example, when designing a building, architects use parallel lines to ensure walls are aligned and structures are stable. If two walls are meant to be parallel, they must have the same slope. If one wall passes through points ( − 6 , 9 ) and ( 7 , − 17 ) , any parallel wall must have the same slope, which we calculated to be − 2 . Therefore, the equation of the parallel wall would be in the form y = − 2 x + b , where b is the y-intercept, determining the wall's position.
The equation that represents a line parallel to the one passing through the points (-6, 9) and (7, -17) is y = − 2 x + 13 . Therefore, the correct answer is D.
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