Solve the following pai of linear equation by elimination methods :
`3x + 4y = 10 and 2x - 2y = 2`.

Answer

To solve the pair of linear equations \(3x + 4y = 10\) and \(2x - 2y = 2\) using the elimination method, follow these steps:

### Step 1: Write down the equations
The given equations are:
1. \(3x + 4y = 10\) (Equation 1)
2. \(2x - 2y = 2\) (Equation 2)

### Step 2: Make the coefficients of \(y\) the same
To eliminate \(y\), we can multiply Equation 2 by 2, so that the coefficients of \(y\) in both equations become equal:

\[ 2 \cdot (2x - 2y) = 2 \cdot 2 \]

This gives us:

\[ 4x - 4y = 4 \quad \text{(Equation 3)} \]

### Step 3: Write the modified equations
Now we have:
1. \(3x + 4y = 10\) (Equation 1)
2. \(4x - 4y = 4\) (Equation 3)

### Step 4: Add the equations to eliminate \(y\)
Now, we add Equation 1 and Equation 3:

\[ (3x + 4y) + (4x - 4y) = 10 + 4 \]

This simplifies to:

\[ 3x + 4y + 4x - 4y = 14 \]

The \(4y\) and \(-4y\) cancel out:

\[ 7x = 14 \]

### Step 5: Solve for \(x\)
Now, divide both sides by 7:

\[ x = \frac{14}{7} = 2 \]

### Step 6: Substitute \(x\) back into one of the original equations
Now that we have \(x = 2\), we can substitute this value back into Equation 1 to find \(y\):

\[ 3(2) + 4y = 10 \]

This simplifies to:

\[ 6 + 4y = 10 \]

### Step 7: Solve for \(y\)
Now, isolate \(4y\):

\[ 4y = 10 - 6 \]
\[ 4y = 4 \]

Now, divide both sides by 4:

\[ y = \frac{4}{4} = 1 \]

### Final Solution
The solution to the system of equations is:
\[ x = 2, \quad y = 1 \]