The straight line passing through the point of intersection of the straight line `x+2y-10=0 and 2x+y+5=0` is

Options

A. `5x-4y=0`
B. ` 5x+4y=0`
C. `4x- 5y=0`
D. ` 4x+ 5y=0`

Answer

To find the straight line passing through the point of intersection of the lines given by the equations \( x + 2y - 10 = 0 \) and \( 2x + y + 5 = 0 \), we will follow these steps:

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

We have the equations:
1. \( x + 2y - 10 = 0 \) (Equation 1)
2. \( 2x + y + 5 = 0 \) (Equation 2)

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

### Step 2: Express one variable in terms of the other.

From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = 10 - 2y \]

### Step 3: Substitute this expression into the second equation.

Now, substitute \( x = 10 - 2y \) into Equation 2:
\[ 2(10 - 2y) + y + 5 = 0 \]
This simplifies to:
\[ 20 - 4y + y + 5 = 0 \]
\[ 25 - 3y = 0 \]

### Step 4: Solve for \( y \).

Rearranging gives:
\[ 3y = 25 \implies y = \frac{25}{3} \]

### Step 5: Substitute \( y \) back to find \( x \).

Now substitute \( y = \frac{25}{3} \) back into the expression for \( x \):
\[ x = 10 - 2\left(\frac{25}{3}\right) = 10 - \frac{50}{3} = \frac{30}{3} - \frac{50}{3} = -\frac{20}{3} \]

### Step 6: Point of intersection.

Thus, the point of intersection is:
\[ \left(-\frac{20}{3}, \frac{25}{3}\right) \]

### Step 7: Check which line passes through this point.

Now we will check which of the given options passes through the point \(\left(-\frac{20}{3}, \frac{25}{3}\right)\).

1. **Option 1:** \( 5x - 4y = 0 \)
\[ 5\left(-\frac{20}{3}\right) - 4\left(\frac{25}{3}\right) = -\frac{100}{3} - \frac{100}{3} = -\frac{200}{3} \neq 0 \]

2. **Option 2:** \( 5x + 4y = 0 \)
\[ 5\left(-\frac{20}{3}\right) + 4\left(\frac{25}{3}\right) = -\frac{100}{3} + \frac{100}{3} = 0 \]
This is true, so this option is valid.

3. **Option 3:** \( 4x - 5y = 0 \)
\[ 4\left(-\frac{20}{3}\right) - 5\left(\frac{25}{3}\right) = -\frac{80}{3} - \frac{125}{3} = -\frac{205}{3} \neq 0 \]

4. **Option 4:** \( 4x + 5y = 0 \)
\[ 4\left(-\frac{20}{3}\right) + 5\left(\frac{25}{3}\right) = -\frac{80}{3} + \frac{125}{3} = \frac{45}{3} \neq 0 \]

### Conclusion:

The only line that passes through the point of intersection is:
\[ \boxed{5x + 4y = 0} \]