The line 5x + 4y = 0 passes through the point of intersection of straight lines (1) x+2y-10 = 0, 2x + y =-5

Explanation

To determine whether the line \(5x + 4y = 0\) passes through the point of intersection of the lines \(x + 2y - 10 = 0\) and \(2x + y = -5\), 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. \(2x + y = -5\) (Equation 2)

To find the intersection, we can solve these equations simultaneously. First, we can express \(y\) from Equation 2:

\[ y = -5 - 2x \quad \text{(from Equation 2)} \]

### Step 2: Substitute \(y\) in Equation 1.

Now, substitute \(y\) in Equation 1:

\[ x + 2(-5 - 2x) - 10 = 0 \]

This simplifies to:

\[ x - 10 - 4x - 10 = 0 \]

Combining like terms:

\[ -3x - 20 = 0 \]

### Step 3: Solve for \(x\).

Now, solve for \(x\):

\[ -3x = 20 \implies x = -\frac{20}{3} \]

### Step 4: Substitute \(x\) back to find \(y\).

Now, substitute \(x = -\frac{20}{3}\) back into the expression for \(y\):

\[ y = -5 - 2\left(-\frac{20}{3}\right) \]

Calculating this gives:

\[ y = -5 + \frac{40}{3} = -\frac{15}{3} + \frac{40}{3} = \frac{25}{3} \]

### Step 5: Point of intersection.

Thus, the point of intersection of the two lines is:

\[ \left(-\frac{20}{3}, \frac{25}{3}\right) \]

### Step 6: Check if this point satisfies the line \(5x + 4y = 0\).

Now, we need to check if this point satisfies the equation \(5x + 4y = 0\):

Substituting \(x = -\frac{20}{3}\) and \(y = \frac{25}{3}\):

\[ 5\left(-\frac{20}{3}\right) + 4\left(\frac{25}{3}\right) = -\frac{100}{3} + \frac{100}{3} = 0 \]

### Conclusion:

Since the left-hand side equals the right-hand side (0), we conclude that the line \(5x + 4y = 0\) does indeed pass through the point of intersection of the two lines.

### Final Answer:

The statement is **True**.

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