Find a linear equation whose graph is the straight line with the given properties.

through (1/3, -1) and parallel to the line 3*x* − 4*y* = 2

To find a linear equation whose graph is a straight line passing through a given point and is parallel to a given line, we can follow these steps:

### 1. Determine the Slope of the Given Line
The given line equation is:

\[
3x - 4y = 2
\]

First, let's rewrite it in the slope-intercept form (\(y = mx + b\)) to determine its slope. Starting from the given equation:

\[
3x - 4y = 2
\]

We solve for \(y\):

\[
-4y = -3x + 2
\]

\[
y = \frac{3}{4}x - \frac{1}{2}
\]

From this equation, we can see that the slope (\(m\)) of the given line is:

\[
m = \frac{3}{4}
\]

### 2. Use the Slope for the Parallel Line
Lines that are parallel have the same slope. Therefore, the slope of the line we need to find is also \( \frac{3}{4} \).

### 3. Find the Equation of the Line Using Point-Slope Form
We know the line passes through the point \( \left(\frac{1}{3}, -1\right) \) and has a slope of \( \frac{3}{4} \). We can use the point-slope form of the line equation:

\[
y - y_1 = m(x - x_1)
\]

Substituting the given point \(\left(\frac{1}{3}, -1\right)\) and slope \( \frac{3}{4} \):

\[
y - (-1) = \frac{3}{4}\left(x - \frac{1}{3}\right)
\]

\[
y + 1 = \frac{3}{4}x - \frac{3}{4} \cdot \frac{1}{3}
\]

\[
y + 1 = \frac{3}{4}x - \frac{1}{4}
\]

### 4. Solve for \(y\) to Get the Slope-Intercept Form
Now, let's solve for \(y\) to put the equation into slope-intercept form:

\[
y = \frac{3}{4}x - \frac{1}{4} - 1
\]

\[
y = \frac{3}{4}x - \frac{1}{4} - \frac{4}{4}
\]

\[
y = \frac{3}{4}x - \frac{5}{4}
\]

### Final Equation
The linear equation of the line passing through \( \left(\frac{1}{3}, -1\right) \) and parallel to the line \( 3x - 4y = 2 \) is:

\[
y = \frac{3}{4}x - \frac{5}{4}
\]

Would you like further details or have any questions about this solution?

Here are some related questions:

1. How can we determine if a point lies on a given line using the line's equation?
2. What is the significance of the y-intercept in the slope-intercept form of a line?
3. How would you find the distance between two parallel lines?
4. How can you determine if two lines are perpendicular by their slopes?
5. What are the methods to find the intersection point of two lines?

**Tip:** Remember, when two lines are parallel, their slopes are equal. This property helps in finding equations of parallel lines easily.

Learn how to find a linear equation parallel to 3x - 4y = 2 that passes through the point (1/3, -1). This problem involves determining slopes, using the point-slope form of a line, and understanding properties of parallel lines in geometry and algebra. Suitable for high school students.

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