Which equation represents a line that is parallel to the line that passes through the points $(-6,9)$ and $(7,-17)$?
A. $y=2 x+13$
B. $y=-2 x+13$
C. $y=\frac{1}{2} x+13$
D. $y=-\frac{1}{2} x+13
Answer (1)
Calculate the slope of the line passing through the points ( − 6 , 9 ) and ( 7 , − 17 ) using the slope formula: m = x 2 − x 1 y 2 − y 1 = 7 − ( − 6 ) − 17 − 9 = − 2 .
Identify the equation from the given options that has the same slope as the calculated slope.
The equation y = − 2 x + 13 has a slope of − 2 .
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 .
Explanation
Understanding the Problem We are given two points, ( − 6 , 9 ) and ( 7 , − 17 ) , and we need to find the equation of a line that is parallel to the line passing through these points. Parallel lines have the same slope. The given options are in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept.
Finding the Slope First, we need to calculate the slope of the line passing through the points ( − 6 , 9 ) and ( 7 , − 17 ) . The slope formula is given by: 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.
Calculating the Slope Plugging in the coordinates of the given points, we have: 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 .
Identifying the Parallel Line Now, we need to find the equation of a line from the given options that has the same slope as the calculated slope, which is − 2 . Looking at the options: A. y = 2 x + 13 (slope is 2) B. y = − 2 x + 13 (slope is -2) C. y = 2 1 x + 13 (slope is 2 1 )
D. y = − 2 1 x + 13 (slope is − 2 1 )
Final Answer The equation y = − 2 x + 13 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 line y = − 2 x + 13 is parallel to the line passing through the given points.
Conclusion The equation that represents a line parallel to the line passing through the points ( − 6 , 9 ) and ( 7 , − 17 ) is y = − 2 x + 13 .
Examples
Understanding parallel lines is crucial in various real-world applications, such as architecture and urban planning. For example, when designing buildings, architects use parallel lines to ensure that walls and floors are aligned correctly. In urban planning, parallel streets can help to create organized and efficient transportation systems. The concept of slope is also essential in these applications, as it determines the steepness of roads and the stability of structures. By understanding these mathematical concepts, we can create safer and more functional environments.
