Determine if the lines $y=2 x+5$ and $4 x-2 y=10$ are parallel, perpendicular, or neither.
Answer (1)
Determine the slope of the first line, y = 2 x + 5 , which is m 1 = 2 .
Convert the second equation, 4 x − 2 y = 10 , to slope-intercept form, resulting in y = 2 x − 5 , and identify its slope as m 2 = 2 .
Compare the slopes: since m 1 = m 2 = 2 , the lines are parallel.
Conclude that the lines are parallel because their slopes are equal. parallel
Explanation
Analyze the problem and given data We are given two lines:
Line 1: y = 2 x + 5 Line 2: 4 x − 2 y = 10
We need to determine if these lines are parallel, perpendicular, or neither. To do this, we will compare their slopes.
Find the slope of the first line The first line is already in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept. For Line 1, the slope m 1 = 2 .
Find the slope of the second line We need to rewrite the second equation in slope-intercept form to find its slope. Starting with 4 x − 2 y = 10 , we can isolate y :
Subtract 4 x from both sides: − 2 y = − 4 x + 10 Divide both sides by − 2 : y = 2 x − 5
So, Line 2 in slope-intercept form is y = 2 x − 5 . The slope of Line 2 is m 2 = 2 .
Compare the slopes Now we compare the slopes. We have m 1 = 2 and m 2 = 2 . Since m 1 = m 2 , the lines are parallel.
To check if they are perpendicular, we would need to see if m 1 ⋅ m 2 = − 1 . In this case, 2 ⋅ 2 = 4 , which is not equal to − 1 . Therefore, the lines are not perpendicular.
Conclusion Since the slopes are equal, the lines are parallel. The final answer is that the lines are parallel.
Examples
Understanding whether lines are parallel or perpendicular is crucial in architecture and construction. For example, when designing a building, architects need to ensure that walls are either parallel to each other for stability or perpendicular for creating right angles in rooms. The principles of slope and linear equations are fundamental in ensuring the structural integrity and functionality of buildings.
