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 (1)
Determine the slope of the given line: m = − 2 .
The parallel line has the same slope and passes through ( − 6 , 10 ) . Use the point-slope form to find the equation: y = − 2 x − 2 .
Set y = 0 to find the x -intercept: 0 = − 2 x − 2 .
Solve for x : x = − 1 , so the ordered pair is ( − 1 , 0 ) .
Explanation
Understanding the Problem The problem asks us to find the point on the x -axis that lies on a line parallel to the given line y = − 2 x − 5 and passing through the point ( − 6 , 10 ) . Since the point is on the x -axis, its y -coordinate is 0. So, we are looking for a point of the form ( x , 0 ) .
Finding the Slope First, we need to find the slope of the given line. The equation y = − 2 x − 5 is in slope-intercept form, y = m x + b , where m is the slope and b is the y -intercept. In this case, the slope of the given line is m = − 2 .
Slope of Parallel Line Since the line we are looking for is parallel to the given line, it has the same slope. Therefore, the slope of the parallel line is also − 2 .
Equation of Parallel Line Now we can use the point-slope form of a line to find the equation of the parallel line passing through the point ( − 6 , 10 ) . The point-slope form is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the given point and m is the slope. Plugging in the values, we get:
y − 10 = − 2 ( x − ( − 6 )) y − 10 = − 2 ( x + 6 ) y − 10 = − 2 x − 12 y = − 2 x − 12 + 10 y = − 2 x − 2
Finding the x-coordinate We want to find the point on the x -axis, which means y = 0 . Substitute y = 0 into the equation of the parallel line:
0 = − 2 x − 2 2 x = − 2 x = − 1
Final Answer Therefore, the ordered pair for the point on the x -axis is ( − 1 , 0 ) .
Examples
Understanding parallel lines is crucial in architecture and design. For instance, when designing a building, architects use parallel lines to ensure walls are aligned and structures are stable. The concept of finding a line parallel to another and passing through a specific point helps in mapping out the layout and ensuring symmetry and balance in the design. This also applies to urban planning, where streets and buildings are often arranged in parallel patterns to optimize space and traffic flow. The ability to calculate and visualize these relationships is fundamental in creating functional and aesthetically pleasing environments.
