Solve for x and y :
2x + 3y = 46,
3x + 5y = 74

Answer

To solve the equations \(2x + 3y = 46\) and \(3x + 5y = 74\) using the elimination method, follow these steps:

### Step 1: Write down the equations
We have the following two equations:
1. \(2x + 3y = 46\) (Equation 1)
2. \(3x + 5y = 74\) (Equation 2)

### Step 2: Make the coefficients of \(x\) or \(y\) equal
To eliminate \(x\), we can multiply Equation 1 by 3 and Equation 2 by 2. This will give us:
- \(3(2x + 3y) = 3(46)\) → \(6x + 9y = 138\) (Equation 3)
- \(2(3x + 5y) = 2(74)\) → \(6x + 10y = 148\) (Equation 4)

### Step 3: Subtract the equations
Now, subtract Equation 3 from Equation 4:
\[ (6x + 10y) - (6x + 9y) = 148 - 138 \]
This simplifies to:
\[ y = 10 \]

### Step 4: Substitute the value of \(y\) back into one of the original equations
Now that we have \(y\), substitute \(y = 10\) into Equation 1:
\[ 2x + 3(10) = 46 \]
This simplifies to:
\[ 2x + 30 = 46 \]

### Step 5: Solve for \(x\)
Now, isolate \(x\):
\[ 2x = 46 - 30 \]
\[ 2x = 16 \]
\[ x = \frac{16}{2} = 8 \]

### Final Solution
Thus, the solution is:
\[ x = 8, \quad y = 10 \]