Question

Find an equation of the line through the point (5,2) that is parallel to the line 6y=5-4x

Answer 1

To find the equation of the line through the point (5, 2) that is parallel to the line given by the equation \(6y = 5 - 4x\), we need to first determine the slope of the line described by that equation.

1. **Rewrite the equation in slope-intercept form (y = mx + b):**
\[
6y = 5 - 4x
\]
Divide both sides by 6:
\[
y = \frac{5}{6} - \frac{4}{6}x
\]
This simplifies to:
\[
y = -\frac{2}{3}x + \frac{5}{6}
\]

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

2. **Since we want a line that is parallel to this one, it will have the same slope:**
\[
m = -\frac{2}{3}
\]

3. **Use the point-slope form of the line equation:**
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is a point on the line.

Here, \(m = -\frac{2}{3}\) and the point is \((5, 2)\), so we substitute into the formula:
\[
y - 2 = -\frac{2}{3}(x - 5)
\]

4. **Distribute and simplify:**
\[
y - 2 = -\frac{2}{3}x + \frac{10}{3}
\]
Now, add 2 (or \( \frac{6}{3} \)) to both sides to get \(y\) by itself:
\[
y = -\frac{2}{3}x + \frac{10}{3} + \frac{6}{3}
\]
\[
y = -\frac{2}{3}x + \frac{16}{3}
\]

5. **Final equation of the line:**
The equation of the line through the point \((5, 2)\) that is parallel to the line \(6y = 5 - 4x\) is:
\[
y = -\frac{2}{3}x + \frac{16}{3}
\]

This is the required equation.