Which equation is the equation of a line that is perpendicular to [tex]x+2 y=16[/tex]?
A. [tex]y=-\frac{1}{2} x+6[/tex]
B. [tex]y=\frac{1}{2} x-3[/tex]
C. [tex]y=2 x-2[/tex]
D. [tex]y=-2 x+8[/tex]
Answer (2)
Rewrite the given equation x + 2 y = 16 in slope-intercept form to find its slope: y = − 2 1 x + 8 , so the slope is − 2 1 .
Calculate the negative reciprocal of the slope to find the slope of the perpendicular line: 2 .
Identify the equation among the options with a slope of 2 .
The equation of the line perpendicular to x + 2 y = 16 is y = 2 x − 2 , so the answer is y = 2 x − 2 .
Explanation
Understanding the Problem We are given the equation of a line x + 2 y = 16 and asked to find which of the four given equations represents a line perpendicular to it. To solve this, we need to find the slope of the given line and then determine the negative reciprocal of that slope, which will be the slope of any line perpendicular to it.
Finding the Slope of the Given Line First, let's rewrite the given equation x + 2 y = 16 in slope-intercept form ( y = m x + b ), where m is the slope and b is the y-intercept. Subtract x from both sides: 2 y = − x + 16 . Divide both sides by 2: y = − 2 1 x + 8 . So, the slope of the given line is − 2 1 .
Finding the Slope of the Perpendicular Line The slope of a line perpendicular to the given line is the negative reciprocal of − 2 1 . The negative reciprocal is found by flipping the fraction and changing the sign. So, the negative reciprocal of − 2 1 is 2 . Therefore, we are looking for a line with a slope of 2 .
Identifying Slopes of the Given Options Now, let's examine the four given equations and identify their slopes:
y = − 2 1 x + 6 has a slope of − 2 1 .
y = 2 1 x − 3 has a slope of 2 1 .
y = 2 x − 2 has a slope of 2 .
y = − 2 x + 8 has a slope of − 2 .
Conclusion Comparing the slopes, we see that the equation y = 2 x − 2 has a slope of 2 , which is the negative reciprocal of the slope of the given line. Therefore, the equation of the line perpendicular to x + 2 y = 16 is y = 2 x − 2 .
Examples
Understanding perpendicular lines is crucial in architecture and construction. When designing buildings, architects need to ensure that walls are perpendicular to the ground for stability. This involves calculating slopes and using the concept of negative reciprocals to guarantee right angles. For example, if a roof has a slope of − 3 1 , the supporting beam must have a slope of 3 to ensure it's perfectly perpendicular, providing maximum support.
The slope of the line represented by x + 2 y = 16 is − 2 1 . The negative reciprocal gives us a slope of 2 , which matches the equation in Option C: y = 2 x − 2 . Thus, the line perpendicular to the given line is y = 2 x − 2 .
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