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 \(43x + 31y = 105\) for \(x + y\), we can follow these steps:

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

### Step 2: Add the two equations
We will add Equation 1 and Equation 2 together:
\[ (31x + 43y) + (43x + 31y) = 117 + 105 \]
This simplifies to:
\[ 31x + 43x + 43y + 31y = 222 \]
Combining like terms gives us:
\[ 74x + 74y = 222 \]

### Step 3: Factor out the common term
We can factor out \(74\) from the left side:
\[ 74(x + y) = 222 \]

### Step 4: Solve for \(x + y\)
Now, we divide both sides by \(74\):
\[ x + y = \frac{222}{74} \]
To simplify \(\frac{222}{74}\), we can divide both the numerator and the denominator by \(2\):
\[ x + y = \frac{111}{37} \]

### Step 5: Simplify the fraction
Calculating \(\frac{111}{37}\) gives us approximately \(3\) (since \(37 \times 3 = 111\)).
Thus, we can conclude:
\[ x + y = 3 \]

### Final Answer
The value of \(x + y\) is \(3\).

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