Question

Find an equation of the line that is perpendicular to the line 4x + 6y + 5 = 0

Answer 1

To find the equation of a line that is perpendicular to the line given by the equation \(4x + 6y + 5 = 0\), we start by determining the slope of the given line.

1. **Rewrite the equation in slope-intercept form \((y = mx + b)\)**:

Starting with the given equation:
\[
4x + 6y + 5 = 0
\]
Solve for \(y\):
\[
6y = -4x - 5
\]
\[
y = -\frac{4}{6}x - \frac{5}{6}
\]
Simplifying \(-\frac{4}{6}\) gives:
\[
y = -\frac{2}{3}x - \frac{5}{6}
\]

From this equation, we see that the slope \(m\) of the given line is \(-\frac{2}{3}\).

2. **Find the slope of the perpendicular line**:

The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, we find the negative reciprocal of \(-\frac{2}{3}\):
\[
m_{\text{perpendicular}} = \frac{3}{2}
\]

3. **Write the equation of the perpendicular line**:

Let's assume we want to find the equation of the perpendicular line that passes through a specific point \((x_1, y_1)\). The point can be any point where you want the perpendicular line to pass through. For instance, if you choose the point \((0, 0)\) (the origin), then the equation of the line in point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]
Substituting \(m = \frac{3}{2}\) and the point \((0, 0)\):
\[
y - 0 = \frac{3}{2}(x - 0)
\]
Simplifying this, we get:
\[
y = \frac{3}{2}x
\]

4. **Final result**:

The equation of the line that is perpendicular to \(4x + 6y + 5 = 0\) and passes through the origin is:
\[
y = \frac{3}{2}x
\]

If you need the perpendicular line to pass through a different point, you can substitute that point into the point-slope form.