Find the equation of a line passing through origin and parallel to the line `3x-5y+2=0`.

Answer

To find the equation of a line passing through the origin and parallel to the line given by the equation \(3x - 5y + 2 = 0\), we can follow these steps:

### Step 1: Identify the slope of the given line
To find the slope of the line \(3x - 5y + 2 = 0\), we first rewrite it in the slope-intercept form \(y = mx + c\), where \(m\) is the slope.

Starting with:
\[ 3x - 5y + 2 = 0 \]

Rearranging gives:
\[ -5y = -3x - 2 \]
\[ y = \frac{3}{5}x + \frac{2}{5} \]

From this, we can see that the slope \(m\) of the line is:
\[ m = \frac{3}{5} \]

### Step 2: Use the slope to find the equation of the new line
Since we want to find a line that is parallel to the given line, it will have the same slope. The new line passes through the origin (0, 0), so we can use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the point through which the line passes.

Substituting \(m = \frac{3}{5}\) and the point \((0, 0)\):
\[ y - 0 = \frac{3}{5}(x - 0) \]
This simplifies to:
\[ y = \frac{3}{5}x \]

### Step 3: Convert to standard form
To express the equation in standard form \(Ax + By + C = 0\), we can rearrange the equation:
\[ \frac{3}{5}x - y = 0 \]

Multiplying through by 5 to eliminate the fraction gives:
\[ 3x - 5y = 0 \]

### Final Answer
Thus, the equation of the line passing through the origin and parallel to the line \(3x - 5y + 2 = 0\) is:
\[ 3x - 5y = 0 \]

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