Find the equation of the straight line passing through the intersection of the lines `x-2y=1` and `x+3y=2` and parallel to `3x+4y=0.`

Explanation

To find the equation of the straight line passing through the intersection of the lines \( x - 2y = 1 \) and \( x + 3y = 2 \), and parallel to \( 3x + 4y = 0 \), we will follow these steps:

### Step 1: Find the intersection point of the two lines.

We have the equations:

1. \( x - 2y = 1 \) (Equation 1)
2. \( x + 3y = 2 \) (Equation 2)

From Equation 1, we can express \( x \) in terms of \( y \):

\[ x = 2y + 1 \]

Now, substitute this expression for \( x \) into Equation 2:

\[ (2y + 1) + 3y = 2 \]

Combine like terms:

\[ 5y + 1 = 2 \]

Now, isolate \( y \):

\[ 5y = 2 - 1 \]
\[ 5y = 1 \]
\[ y = \frac{1}{5} \]

Now, substitute \( y = \frac{1}{5} \) back into the expression for \( x \):

\[ x = 2\left(\frac{1}{5}\right) + 1 = \frac{2}{5} + 1 = \frac{2}{5} + \frac{5}{5} = \frac{7}{5} \]

Thus, the intersection point is:

\[ \left( \frac{7}{5}, \frac{1}{5} \right) \]

### Step 2: Determine the slope of the line parallel to \( 3x + 4y = 0 \).

To find the slope of the line \( 3x + 4y = 0 \), we can rewrite it in slope-intercept form \( y = mx + c \):

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

From this, we see that the slope \( m \) is:

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

### Step 3: Use the point-slope form to find the equation of the new line.

We will use the point-slope form of the equation of a line, which is given by:

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

Where \( (x_1, y_1) \) is the point \( \left( \frac{7}{5}, \frac{1}{5} \right) \) and \( m = -\frac{3}{4} \).

Substituting these values into the point-slope form:

\[ y - \frac{1}{5} = -\frac{3}{4}\left(x - \frac{7}{5}\right) \]

### Step 4: Simplify the equation.

Multiply both sides by 4 to eliminate the fraction:

\[ 4\left(y - \frac{1}{5}\right) = -3\left(x - \frac{7}{5}\right) \]

This simplifies to:

\[ 4y - \frac{4}{5} = -3x + \frac{21}{5} \]

Now, rearranging gives:

\[ 3x + 4y = \frac{21}{5} + \frac{4}{5} \]

Combine the constants:

\[ 3x + 4y = \frac{25}{5} \]
\[ 3x + 4y = 5 \]

### Final Answer:

The equation of the straight line is:

\[ 3x + 4y = 5 \]