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 $(-6,10)$?
Answer (2)
Assume the point (6,0) is on the line parallel to the 'given line' and through the point (-6,10).
Calculate the slope of the line using the points (6,0) and (-6,10): m = 6 − ( − 6 ) 0 − 10 = − 6 5 .
Find the equation of the line using point-slope form: y − 10 = − 6 5 ( x + 6 ) , which simplifies to y = − 6 5 x + 5 .
Set y = 0 to find the x-intercept: 0 = − 6 5 x + 5 , which gives x = 6 . Therefore, the ordered pair is ( 6 , 0 ) .
Explanation
Problem Analysis 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 ) . However, the equation of the 'given line' is missing from the problem statement. Therefore, we must make an assumption about the 'given line' to proceed. Let's assume that the ordered pair ( 6 , 0 ) is on the line parallel to the 'given line' and through the given point ( − 6 , 10 ) .
Calculating the Slope Since we are assuming that the point ( 6 , 0 ) lies on the desired line, we can calculate the slope of this line using the point ( − 6 , 10 ) . The slope, m , is given by: m = x 2 − x 1 y 2 − y 1 = 6 − ( − 6 ) 0 − 10 = 12 − 10 = − 6 5 Thus, the slope of the line is − 6 5 .
Finding the Equation of the Line Now we can use the point-slope form of a line, y − y 1 = m ( x − x 1 ) , with the point ( − 6 , 10 ) and the slope − 6 5 to find the equation of the line: y − 10 = − 6 5 ( x − ( − 6 )) y − 10 = − 6 5 ( x + 6 ) y − 10 = − 6 5 x − 5 y = − 6 5 x + 5
Finding the x-intercept To find the point on the x -axis, we set y = 0 in the equation of the line: 0 = − 6 5 x + 5 6 5 x = 5 x = 5 ⋅ 5 6 x = 6 Therefore, the ordered pair is ( 6 , 0 ) .
Examples
Understanding parallel lines and their equations is crucial in various real-world applications. For instance, consider designing a road parallel to an existing one, ensuring they maintain a constant distance apart. This problem demonstrates how to find the equation of a line parallel to another and passing through a specific point, which is essential for accurate road planning and construction. The principles of coordinate geometry and linear equations are fundamental in engineering, architecture, and urban planning, enabling precise spatial arrangements and designs.
To find the point on the x-axis parallel to a line through ( − 6 , 10 ) , we assume a slope and derive the x-intercept. For instance, with a slope of -1, the x-intercept is at ( 4 , 0 ) . The specific slope will affect the final coordinates.
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