Solve : 65x -33y = 97 and 33x - 65y = 1

Answer

To solve the simultaneous equations \(65x - 33y = 97\) and \(33x - 65y = 1\), we can follow these steps:

### Step 1: Write down the equations
We have the following two equations:
1. \(65x - 33y = 97\) (Equation 1)
2. \(33x - 65y = 1\) (Equation 2)

### Step 2: Rearrange Equation 2 to express \(x\) in terms of \(y\)
From Equation 2, we can isolate \(33x\):
\[ 33x = 1 + 65y \]
Now, divide both sides by 33 to find \(x\):
\[ x = \frac{1 + 65y}{33} \]

### Step 3: Substitute \(x\) into Equation 1
Now substitute the expression for \(x\) from Step 2 into Equation 1:
\[ 65\left(\frac{1 + 65y}{33}\right) - 33y = 97 \]

### Step 4: Simplify the equation
Multiply through by 33 to eliminate the fraction:
\[ 65(1 + 65y) - 33 \cdot 33y = 97 \cdot 33 \]
This simplifies to:
\[ 65 + 4225y - 1089y = 3201 \]

### Step 5: Combine like terms
Combine the \(y\) terms:
\[ 65 + (4225 - 1089)y = 3201 \]
This simplifies to:
\[ 65 + 3136y = 3201 \]

### Step 6: Isolate \(y\)
Subtract 65 from both sides:
\[ 3136y = 3201 - 65 \]
\[ 3136y = 3136 \]
Now divide by 3136:
\[ y = 1 \]

### Step 7: Substitute \(y\) back to find \(x\)
Now that we have \(y\), substitute it back into the expression for \(x\):
\[ x = \frac{1 + 65 \cdot 1}{33} \]
\[ x = \frac{66}{33} = 2 \]

### Final Solution
Thus, the solution to the simultaneous equations is:
\[ x = 2, \quad y = 1 \]