Parallel/Perpendicular Through Point
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What is an equation of the line that passes through the point $(4,0)$ and is perpendicular to the line $4 x-3 y=15$ ?
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Answer
The equation of the line that passes through the point \((4,0)\) and is perpendicular to the line \(4x - 3y = 15\) is \(\boxed{y = -\frac{3}{4}x + 3}\).
Step 1 ：The equation of a line in the slope-intercept form is given by \(y = mx + c\), where \(m\) is the slope of the line and \(c\) is the y-intercept.
Step 2 ：The slope of a line given by the equation \(ax + by = c\) is \(-a/b\). Therefore, the slope of the line \(4x - 3y = 15\) is \(4/3\).
Step 3 ：The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. Therefore, the slope of the line we are looking for, which is perpendicular to the line \(4x - 3y = 15\), is \(-3/4\).
Step 4 ：We also know that this line passes through the point \((4,0)\). We can substitute these values into the equation \(y = mx + c\) to find the y-intercept \(c\).
Step 5 ：\(m = -0.75\)
Step 6 ：\(c = 3.0\)
Step 7 ：Now that we have the slope and the y-intercept, we can write the equation of the line in the slope-intercept form.
Step 8 ：The equation of the line that passes through the point \((4,0)\) and is perpendicular to the line \(4x - 3y = 15\) is \(\boxed{y = -\frac{3}{4}x + 3}\).
