Find the equation of the straight line passing through origin and the point of intersection of the lines `x + 2y = 7` and `x - y = 4` . 

Answer

To find the equation of the straight line passing through the origin and the point of intersection of the lines \( x + 2y = 7 \) and \( x - y = 4 \), we can follow these steps:

### Step 1: Find the Point of Intersection of the Two Lines
We need to solve the equations \( x + 2y = 7 \) and \( x - y = 4 \) simultaneously.

1. **Write down the equations:**
\[ (1) \quad x + 2y = 7 \]
\[ (2) \quad x - y = 4 \]

2. **Use the method of elimination:**
- From equation (2), express \( x \) in terms of \( y \):
\[ x = y + 4 \]
- Substitute \( x \) in equation (1):
\[ (y + 4) + 2y = 7 \]
- Simplifying gives:
\[ 3y + 4 = 7 \]
\[ 3y = 3 \]
\[ y = 1 \]

3. **Substitute \( y \) back to find \( x \):**
- Using \( y = 1 \) in equation (2):
\[ x - 1 = 4 \implies x = 5 \]
- Thus, the point of intersection is \( (5, 1) \).

### Step 2: Find the Slope of the Line
Now we have two points: the origin \( (0, 0) \) and the point of intersection \( (5, 1) \).

1. **Use the slope formula:**
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
- Let \( (x_1, y_1) = (0, 0) \) and \( (x_2, y_2) = (5, 1) \):
\[ m = \frac{1 - 0}{5 - 0} = \frac{1}{5} \]

### Step 3: Write the Equation of the Line
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \( (x_1, y_1) = (0, 0) \) and \( m = \frac{1}{5} \):
\[ y - 0 = \frac{1}{5}(x - 0) \implies y = \frac{1}{5}x \]

### Step 4: Rearranging the Equation
To express the equation in standard form:
\[ 5y = x \]

### Final Answer
The equation of the straight line passing through the origin and the point of intersection of the given lines is:
\[ x - 5y = 0 \]

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