If 3x+5y =9 and 5x+3y=7, then find the value of x+y.

Answer

To solve the equations \(3x + 5y = 9\) and \(5x + 3y = 7\) and find the value of \(x + y\), we can use the method of substitution or elimination. Here, we will use the elimination method.

### Step 1: Write down the equations
We have:
1. \(3x + 5y = 9\) (Equation 1)
2. \(5x + 3y = 7\) (Equation 2)

### Step 2: Multiply the equations to align coefficients
To eliminate one of the variables, we can multiply Equation 1 by 5 and Equation 2 by 3 to make the coefficients of \(x\) the same.

- Multiply Equation 1 by 5:
\[ 15x + 25y = 45 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by 3:
\[ 15x + 9y = 21 \quad \text{(Equation 4)} \]

### Step 3: Subtract the equations
Now, we will subtract Equation 4 from Equation 3 to eliminate \(x\):
\[ (15x + 25y) - (15x + 9y) = 45 - 21 \]
This simplifies to:
\[ 16y = 24 \]

### Step 4: Solve for \(y\)
Now, divide both sides by 16:
\[ y = \frac{24}{16} = \frac{3}{2} \]

### Step 5: Substitute \(y\) back into one of the original equations
We can substitute \(y = \frac{3}{2}\) back into Equation 1 to find \(x\):
\[ 3x + 5\left(\frac{3}{2}\right) = 9 \]
This simplifies to:
\[ 3x + \frac{15}{2} = 9 \]

### Step 6: Clear the fraction
Multiply the entire equation by 2 to eliminate the fraction:
\[ 6x + 15 = 18 \]

### Step 7: Solve for \(x\)
Now, subtract 15 from both sides:
\[ 6x = 3 \]
Then, divide by 6:
\[ x = \frac{3}{6} = \frac{1}{2} \]

### Step 8: Find \(x + y\)
Now that we have \(x\) and \(y\):
\[ x + y = \frac{1}{2} + \frac{3}{2} = \frac{4}{2} = 2 \]

### Final Answer
Thus, the value of \(x + y\) is \(2\).

---