Find the equation of a line passing through the intersection of the lines `3x-y=1` and `5x+2y=9` and parallel to the line `3x+5y=8`.

Answer

To find the equation of a line passing through the intersection of the lines \(3x - y = 1\) and \(5x + 2y = 9\) and parallel to the line \(3x + 5y = 8\), we can follow these steps:

### Step 1: Find the intersection of the two lines
We have the equations:
1. \(3x - y = 1\) (Equation 1)
2. \(5x + 2y = 9\) (Equation 2)

To find the intersection, we can solve these two equations simultaneously.

From Equation 1, we can express \(y\) in terms of \(x\):
\[ y = 3x - 1 \]

Now, substitute \(y\) in Equation 2:
\[ 5x + 2(3x - 1) = 9 \]
Expanding this gives:
\[ 5x + 6x - 2 = 9 \]
Combining like terms:
\[ 11x - 2 = 9 \]
Adding 2 to both sides:
\[ 11x = 11 \]
Dividing by 11:
\[ x = 1 \]

Now, substitute \(x = 1\) back into the expression for \(y\):
\[ y = 3(1) - 1 = 2 \]

Thus, the intersection point is \((1, 2)\).

### Step 2: Find the slope of the line parallel to \(3x + 5y = 8\)
The given line is \(3x + 5y = 8\). To find its slope, we can rewrite it in slope-intercept form \(y = mx + b\):
\[ 5y = -3x + 8 \]
\[ y = -\frac{3}{5}x + \frac{8}{5} \]
The slope \(m\) of this line is \(-\frac{3}{5}\).

Since we need a line that is parallel to this, it will have the same slope.

### Step 3: Use the point-slope form to find the equation of the new line
We can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the point \((1, 2)\) and \(m = -\frac{3}{5}\).

Substituting these values:
\[ y - 2 = -\frac{3}{5}(x - 1) \]

### Step 4: Simplify the equation
Distributing the slope:
\[ y - 2 = -\frac{3}{5}x + \frac{3}{5} \]
Adding 2 to both sides:
\[ y = -\frac{3}{5}x + \frac{3}{5} + 2 \]
Converting 2 to a fraction:
\[ y = -\frac{3}{5}x + \frac{3}{5} + \frac{10}{5} \]
Combining the constants:
\[ y = -\frac{3}{5}x + \frac{13}{5} \]

### Step 5: Convert to standard form
To convert this to standard form \(Ax + By = C\), we can multiply through by 5 to eliminate the fractions:
\[ 5y = -3x + 13 \]
Rearranging gives:
\[ 3x + 5y = 13 \]

### Final Answer
The equation of the line passing through the intersection of the two lines and parallel to the given line is:
\[ \boxed{3x + 5y = 13} \]