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 slope of the perpendicular line by taking the negative reciprocal of the given line's slope: − − 2 1 1 = 2 .
Examine the given options and identify the line with a slope of 2.
The equation of the line perpendicular to x + 2 y = 16 is y = 2 x − 2 .
Explanation
Find the slope of the given line First, we need to determine the slope of the given line, which is x + 2 y = 16 . To do this, we rewrite the equation in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Rewrite in slope-intercept form Subtract x from both sides of the equation: 2 y = − x + 16 . Then, divide both sides by 2: y = − 2 1 x + 8 . So, the slope of the given line is − 2 1 .
Calculate the slope of the perpendicular line The slope of a line perpendicular to the given line is the negative reciprocal of the slope of the given line. If the slope of the given line is m , then the slope of the perpendicular line is − m 1 . In this case, the slope of the perpendicular line is − − 2 1 1 = 2 .
Find the line with the correct slope Now we need to find which of the given lines has a slope of 2. Let's examine each option:
y = − 2 1 x + 6 : The slope is − 2 1 .
y = 2 1 x − 3 : The slope is 2 1 .
y = 2 x − 2 : The slope is 2.
y = − 2 x + 8 : The slope is -2.
Conclusion The line 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 that is perpendicular to x + 2 y = 16 is y = 2 x − 2 .
Examples
Understanding perpendicular lines is crucial in various real-world applications, such as architecture and construction. For example, when designing a building, architects need to ensure that walls are perpendicular to the ground to maintain structural integrity. Similarly, in road construction, engineers use the concept of perpendicular lines to design intersections and ensure safe traffic flow. The relationship between slopes of perpendicular lines helps in creating accurate and stable structures.
The equation of a line that is perpendicular to x + 2 y = 16 is found to be y = 2 x − 2 , as it has a slope of 2, which is the negative reciprocal of the original line's slope. The original line's slope was calculated to be − 2 1 . Therefore, the answer is C: y = 2 x − 2 .
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