Given the equatins above, what is the value of `x +y` ?

Answer

To solve the given equations and find the value of \( x + y \), we will follow these steps:

### Step 1: Write down the equations
The given equations are:
1. \( 3x = 4y - 13 \) (Equation 1)
2. \( 3y = 31 - 2x \) (Equation 2)

### Step 2: Rearrange the equations
We can rearrange both equations to express them in a standard form.

From Equation 1:
\[ 3x - 4y = -13 \quad \text{(Rearranged Equation 1)} \]

From Equation 2:
\[ 2x + 3y = 31 \quad \text{(Rearranged Equation 2)} \]

### Step 3: Eliminate one variable
To eliminate \( x \), we can multiply the first equation by 2 and the second equation by 3 to make the coefficients of \( x \) the same.

Multiplying the rearranged Equation 1 by 2:
\[ 2(3x - 4y) = 2(-13) \implies 6x - 8y = -26 \quad \text{(Equation 3)} \]

Multiplying the rearranged Equation 2 by 3:
\[ 3(2x + 3y) = 3(31) \implies 6x + 9y = 93 \quad \text{(Equation 4)} \]

### Step 4: Add the equations
Now we can add Equation 3 and Equation 4:
\[ (6x - 8y) + (6x + 9y) = -26 + 93 \]
This simplifies to:
\[ 12x + y = 67 \]

### Step 5: Solve for \( y \)
From the equation \( 12x + y = 67 \), we can express \( y \) in terms of \( x \):
\[ y = 67 - 12x \quad \text{(Equation 5)} \]

### Step 6: Substitute back to find \( x \)
Now we can substitute \( y \) back into one of the original equations to find \( x \). Let's use Equation 1:
\[ 3x = 4(67 - 12x) - 13 \]
Expanding this gives:
\[ 3x = 268 - 48x - 13 \]
Combining like terms:
\[ 3x + 48x = 255 \implies 51x = 255 \implies x = 5 \]

### Step 7: Find \( y \)
Now substitute \( x = 5 \) back into Equation 5 to find \( y \):
\[ y = 67 - 12(5) = 67 - 60 = 7 \]

### Step 8: Calculate \( x + y \)
Finally, we can find \( x + y \):
\[ x + y = 5 + 7 = 12 \]

### Final Answer
The value of \( x + y \) is \( \boxed{12} \).

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