If the graph of the eqations ` x + y = 0 and 5 y + 7 x = 24` intersect at (m, n) then the value of m + n is

Options

A. 2
B. 1
C. 0
D. `-1`

Answer

To find the value of \( m + n \) where the equations \( x + y = 0 \) and \( 5y + 7x = 24 \) intersect at the point \( (m, n) \), we will solve the system of equations step by step.

### Step 1: Write down the equations
We have the following two equations:
1. \( x + y = 0 \)
2. \( 5y + 7x = 24 \)

### Step 2: Solve the first equation for \( y \)
From the first equation, we can express \( y \) in terms of \( x \):
\[ y = -x \]

### Step 3: Substitute \( y \) in the second equation
Now, we will substitute \( y = -x \) into the second equation:
\[ 5(-x) + 7x = 24 \]
This simplifies to:
\[ -5x + 7x = 24 \]

### Step 4: Combine like terms
Combine the \( x \) terms:
\[ 2x = 24 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 2 to find \( x \):
\[ x = \frac{24}{2} = 12 \]

### Step 6: Find \( y \) using the value of \( x \)
Now that we have \( x \), we can find \( y \) using the expression \( y = -x \):
\[ y = -12 \]

### Step 7: Determine the coordinates of the intersection point
Thus, the intersection point \( (m, n) \) is:
\[ (m, n) = (12, -12) \]

### Step 8: Calculate \( m + n \)
Finally, we calculate \( m + n \):
\[ m + n = 12 + (-12) = 0 \]

### Final Answer
The value of \( m + n \) is \( 0 \).

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