2. Calculate the Negative Reciprocal of the Slope: Find the negative reciprocal of the slope, denoted as \(\frac{-1}{m}\). This value represents the slope of the perpendicular line.
3. Identify a Point on the Perpendicular Line: Select a point that lies on the perpendicular line.
You can either use a provided point or choose a convenient one.
4. Use the Point-Slope Form to Write the Equation: Utilize the point-slope form of a line, which is y - y1 = m(x - x1).
Substitute the negative reciprocal of the slope \((\frac{-1}{m})\) for m and the coordinates of the point (x1, y1) into the equation.
5. Simplify the Equation: Rearrange and simplify the equation, moving all terms to one side, to obtain the standard form \(Ax + By = C\).
This form is commonly used for linear equations.
Example:
Given Line: \(y = 2x + 3\)
1. Determine the Slope of the Given Line: The slope of the given line is \(m=2\).
2. Calculate the Negative Reciprocal of the Slope: The negative reciprocal of 2 is \(\frac{-1}{2}\).
3. Identify a Point on the Perpendicular Line: Let's choose the point \((1, 4)\) on the given line.
4. Use the Point-Slope Form to Write the Equation: Applying the point-slope form, we have:
\(y - 4 = (\frac{-1}{2})(x - 1)\)
5. Simplify the Equation: Simplifying, we get:
\(y - 4 = \frac{-1}{2}x + \frac{1}{2}\)
Multiply both sides by 2 to eliminate fractions:
\(2(y - 4) = -x + 1\)
Distribute 2 and rearrange the terms:
\(2y - 8 = -x + 1\)
Move all terms to one side:
\(x + 2y = 9\)
Thus, the equation of the perpendicular line to \(y = 2x + 3\), passing through the point (1, 4), is \(x + 2y = 9\) in standard form.
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