What is the equation of the line that is perpendicular to the given line and has an $x$-intercept of 6?
$y=-\frac{3}{4} x+8$
$y=-\frac{3}{4} x+6$
$y=\frac{4}{3} x-8$
$y=\frac{4}{3} x-6
Answer (2)
Find the slope of the given line: m 1 = − 4 3 .
Determine the slope of the perpendicular line: m 2 = 3 4 .
Use the point-slope form with the point (6, 0): y − 0 = 3 4 ( x − 6 ) .
Rewrite in slope-intercept form: y = 3 4 x − 8 . The final answer is y = 3 4 x − 8 .
Explanation
Find the slope of the given line The given line is in the slope-intercept form y = m x + b , where m represents the slope. In the equation y = − 4 3 x + 8 , the slope of the given line is m 1 = − 4 3 .
Determine 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 m 1 is the slope of the given line and m 2 is the slope of the perpendicular line, then m 2 = − m 1 1 . In this case, m 1 = − 4 3 , so the slope of the perpendicular line is m 2 = − − 4 3 1 = 3 4 .
Use the point-slope form to find the equation The x -intercept of the perpendicular line is 6, which means the line passes through the point (6, 0). We can use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is a point on the line and m is the slope. Plugging in the point (6, 0) and the slope 3 4 , we get y − 0 = 3 4 ( x − 6 ) .
Rewrite the equation in slope-intercept form Now, we rewrite the equation in slope-intercept form, y = m x + b . y = 3 4 ( x − 6 ) = 3 4 x − 3 4 ( 6 ) = 3 4 x − 8. Thus, the equation of the line is y = 3 4 x − 8 .
Final Answer The equation of the line that is perpendicular to the given line y = − 4 3 x + 8 and has an x -intercept of 6 is y = 3 4 x − 8 .
Examples
Understanding perpendicular lines is crucial in architecture and construction. For example, when designing a building, ensuring walls are perpendicular to the ground is essential for stability. If a roof's slope is represented by the line y = − 4 3 x + 8 , a support beam perpendicular to this slope would follow the equation y = 3 4 x − 8 . This ensures the beam provides the necessary support at a right angle, maximizing structural integrity.
The equation of the line that is perpendicular to the given line and has an x -intercept of 6 is y = 3 4 x − 8 . This was found by determining the slope of the original line, calculating the negative reciprocal for the perpendicular slope, and using the x-intercept to find the final equation. Thus, the answer is option y = 3 4 x − 8 .
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