Write an equation for a line that is perpendicular to $2 y-3 x=8$ and goes through $(6,2)$.

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01-11-2026
Mathematics

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Write an equation for a line that is perpendicular to $2 y-3 x=8$ and goes through $(6,2)$.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-01T23:59:39-05:00

Convert the given line equation $2y - 3x = 8$ to slope-intercept form to find its slope $m_1 = \frac{3}{2}$.
Determine the slope of the perpendicular line using the property $m_1 \cdot m_2 = -1$, which gives $m_2 = -\frac{2}{3}$.
Use the point-slope form with the point $(6, 2)$ to derive the equation $y - 2 = -\frac{2}{3}(x - 6)$.
Simplify the final equation to the standard form: $\boxed{2x + 3y = 18}$.

Explanation

Determine the slope of the given line

To find the equation of a line perpendicular to $2y - 3x = 8$ that passes through the point $(6, 2)$, we first identify the slope of the given line. We can rewrite the given equation $2y - 3x = 8$ in the slope-intercept form, $y = mx + b$, where $m$ is the slope.
Adding $3x$ to both sides, we get:
\[2y = 3x + 8\]
Dividing by 2 gives:
\[y = \frac{3}{2}x + 4\]
The slope of the given line is $m_1 = \frac{3}{2}$.

Calculate the slope of the perpendicular line

Two lines are perpendicular if the product of their slopes is $-1$. Let $m_2$ be the slope of the line perpendicular to our given line. Then we have the relation:
\[m_1 \cdot m_2 = -1\]
Substituting $m_1 = \frac{3}{2}$:
\[\frac{3}{2} \cdot m_2 = -1\]
Multiplying both sides by $\frac{2}{3}$, we find the slope of our new line:
\[m_2 = -\frac{2}{3}\]

Find the equation of the perpendicular line

Now that we have the slope $m_2 = -\frac{2}{3}$ and a point $(6, 2)$ that the line passes through, we use the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$:
\[y - 2 = -\frac{2}{3}(x - 6)\]
Distributing the $-\frac{2}{3}$ on the right side:
\[y - 2 = -\frac{2}{3}x + \left(-\frac{2}{3} \cdot -6\right)\]
\[y - 2 = -\frac{2}{3}x + 4\]
Adding 2 to both sides gives the equation in slope-intercept form:
\[y = -\frac{2}{3}x + 6\]
To write this in standard form, we can multiply the entire equation by 3 to eliminate the fraction:
\[3y = -2x + 18\]
Rearranging terms gives:
\[2x + 3y = 18\]
Both forms $y = -\frac{2}{3}x + 6$ and $2x + 3y = 18$ are valid representations of the line.

Examples

In urban planning, streets are often designed to be perpendicular to each other for grid layouts. If one street is represented by a linear equation on a city map, civil engineers calculate the equation of the perpendicular street to determine where to place intersections or install traffic signals at specific coordinates. This ensures organized traffic flow and efficient city design, demonstrating the real-world utility of coordinate geometry.