If 31x+43y=117 and 43x+31y=105 then, the value of x+y is:

Options

A. -3
B. `1/3`
C. `-1/3`
D. 3

Answer

To solve the equations \(31x + 43y = 117\) and \(43x + 31y = 105\) and find the value of \(x + y\), we can follow these steps:

### Step 1: Write down the equations
We have two equations:
1. \(31x + 43y = 117\) (Equation 1)
2. \(43x + 31y = 105\) (Equation 2)

### Step 2: Add the two equations
We will add both equations together:
\[ (31x + 43y) + (43x + 31y) = 117 + 105 \]

### Step 3: Combine like terms
Combining the terms on the left side:
\[ (31x + 43x) + (43y + 31y) = 222 \]
This simplifies to:
\[ 74x + 74y = 222 \]

### Step 4: Factor out the common term
We can factor out 74 from the left side:
\[ 74(x + y) = 222 \]

### Step 5: Solve for \(x + y\)
Now, divide both sides by 74:
\[ x + y = \frac{222}{74} \]

### Step 6: Simplify the fraction
To simplify \(\frac{222}{74}\), we can divide both the numerator and the denominator by 2:
\[ x + y = \frac{111}{37} \]

### Final Answer
Thus, the value of \(x + y\) is \(\frac{111}{37}\).

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