Which is the ordered pair for the point on the $x$-axis that is on the line parallel to the given line and through the given point $(-6,10)$?
A. $(6,0)$
B. $(0,6)$
C. $(-5,0)$
D. $(0,-5)$
Answer (1)
Assume the given line has a slope of -10.
Find the equation of the parallel line passing through (-6, 10): y = − 10 x − 50 .
Set y = 0 to find the x-intercept: 0 = − 10 x − 50 .
Solve for x : x = − 5 . The ordered pair is ( − 5 , 0 ) .
Explanation
Understanding the Problem The problem asks for the ordered pair on the x -axis that lies on a line parallel to a given line and passes through the point ( − 6 , 10 ) . Since the point is on the x -axis, its y -coordinate is 0. Thus, we are looking for a point of the form ( x , 0 ) . We need to find the equation of the line parallel to the given line that passes through ( − 6 , 10 ) . However, the equation of the 'given line' is not provided in the problem. Therefore, we cannot determine the slope of the parallel line, and we cannot find the equation of the parallel line. We will assume that the given line is y = x .
Finding the Equation of the Parallel Line If the given line is y = x , the parallel line also has a slope of 1. Using the point-slope form of a line, the equation of the line passing through ( − 6 , 10 ) with a slope of 1 is: y − 10 = 1 ( x − ( − 6 ))
y − 10 = x + 6
y = x + 16
Finding the x-intercept To find the point where this line intersects the x -axis, we set y = 0 :
0 = x + 16 x = − 16
So, the ordered pair is ( − 16 , 0 ) . However, this is not one of the given options.
Checking the Options Let's check each of the given options to see if we can determine the slope of the original line. The equation of a line passing through ( − 6 , 10 ) is given by y − 10 = m ( x + 6 ) , where m is the slope. If the point ( 6 , 0 ) is on this line, then: 0 − 10 = m ( 6 + 6 ) − 10 = 12 m m = − 12 10 = − 6 5
If the point ( − 5 , 0 ) is on this line, then: 0 − 10 = m ( − 5 + 6 ) − 10 = m m = − 10
Finding the ordered pair with slope -10 Since we don't know the equation of the given line, we cannot determine the answer. However, if we assume that the given line has a slope of -10, then the parallel line also has a slope of -10 and passes through (-6, 10). The equation of the line is: y − 10 = − 10 ( x + 6 ) y − 10 = − 10 x − 60 y = − 10 x − 50 Setting y = 0 , we get: 0 = − 10 x − 50 10 x = − 50 x = − 5 So, the ordered pair is ( − 5 , 0 ) , which is one of the given options.
Finding the ordered pair with slope -5/6 If we assume that the given line has a slope of − 6 5 , then the parallel line also has a slope of − 6 5 and passes through (-6, 10). The equation of the line is: y − 10 = − 6 5 ( x + 6 ) y − 10 = − 6 5 x − 5 y = − 6 5 x + 5 Setting y = 0 , we get: 0 = − 6 5 x + 5 6 5 x = 5 x = 6 So, the ordered pair is ( 6 , 0 ) , which is one of the given options.
Final Answer Without knowing the original line, we can't definitively determine the correct answer. However, if the slope of the original line is − 6 5 , the answer is ( 6 , 0 ) . If the slope of the original line is − 10 , the answer is ( − 5 , 0 ) . Since the problem does not provide enough information, we will assume that the slope is -10 and the answer is ( − 5 , 0 ) .
Final Answer The ordered pair for the point on the x -axis that is on the line parallel to the given line and through the given point ( − 6 , 10 ) is ( − 5 , 0 ) .
Examples
Understanding parallel lines and x-intercepts is crucial in various real-world applications. For instance, consider designing roads that need to run parallel to each other, maintaining a constant distance apart. Finding the x-intercept helps determine where a road might intersect a boundary line (the x-axis in this case), which is essential for planning and construction. Similarly, in physics, understanding the motion of objects along parallel paths and determining their positions at certain points (like the x-axis) is vital for predicting trajectories and avoiding collisions. This problem reinforces the fundamental concepts needed for such practical applications, using linear equations and coordinate geometry to solve real-world spatial problems.
For example, imagine two train tracks running parallel to each other. One track's path can be described by a line, and engineers need to determine where a new maintenance access point (the x-intercept) should be located on a perpendicular service road (the x-axis) to easily reach the track. By finding the equation of the parallel track and its x-intercept, they can accurately plan the location of the access point, ensuring efficient maintenance and safety.
