Geometric Proof: Visualizing Perpendicular Slope Calculations on a Coordinate Grid

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To put these mechanics into practice, let us solve a standard coordinate geometry problem: find the equation of a line perpendicular to $3x + 4y = 12$ that passes through the specific point $(6, -1)$.

Step 1: Extract the Original Slope

Convert the standard form linear equation into slope-intercept form ($y = mx + b$) by isolating $y$:

$$4y = -3x + 12$$

$$y = -\frac{3}{4}x + 3$$

The slope of the initial trajectory ($m_1$) is $-\frac{3}{4}$.

Step 2: Determine the Negative Reciprocal Slope

Apply the perpendicular condition $m1 \cdot m2 = -1$:

$$-\frac{3}{4} \cdot m_2 = -1$$

$$m_2 = \frac{-1}{-\frac{3}{4}} = \frac{4}{3}$$

The new slope is positive $4/3$.

Step 3: Apply Point-Slope Form

Use point-slope form with the known target coordinate $(x1, y1) = (6, -1)$:

$$y - y1 = m(x - x1)$$

$$y - (-1) = \frac{4}{3}(x - 6)$$

$$y + 1 = \frac{4}{3}x - 8$$

Step 4: Execute the y-Intercept Calculation

Subtract $1$ from both sides to finish the y-intercept calculation:

$$y = \frac{4}{3}x - 9$$

To express this result in standard form, multiply every term by $3$ to eliminate fractions:

$$3y = 4x - 27 \implies 4x - 3y = 27$$

Notice the coefficient relationship: the original line was $3x + 4y = 12$, and its orthogonal companion is $4x - 3y = 27$. The coefficients swapped positions, and the addition changed to subtraction.