Solve the linear equations by elimination method
`{:(41x+53y=135),(53x+41y=147):}`

Answer

To solve the system of linear equations using the elimination method, we have the following equations:

1. \( 41x + 53y = 135 \) (Equation 1)
2. \( 53x + 41y = 147 \) (Equation 2)

### Step 1: Multiply the equations to align coefficients
To eliminate one of the variables, we can multiply both equations by suitable numbers so that the coefficients of one variable are the same.

- Multiply Equation 1 by \( 53 \):
\[ 53(41x + 53y) = 53(135) \]
This gives us:
\[ 2153x + 2809y = 7155 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by \( 41 \):
\[ 41(53x + 41y) = 41(147) \]
This gives us:
\[ 2153x + 1681y = 6037 \quad \text{(Equation 4)} \]

### Step 2: Subtract the equations
Now we will subtract Equation 4 from Equation 3 to eliminate \( x \):
\[ (2153x + 2809y) - (2153x + 1681y) = 7155 - 6037 \]
This simplifies to:
\[ 2809y - 1681y = 7155 - 6037 \]
\[ 1128y = 1118 \]

### Step 3: Solve for \( y \)
Now, divide both sides by \( 1128 \):
\[ y = \frac{1118}{1128} = 1 \]

### Step 4: Substitute \( y \) back into one of the original equations
Now that we have \( y \), we can substitute it back into Equation 1 to find \( x \):
\[ 41x + 53(1) = 135 \]
This simplifies to:
\[ 41x + 53 = 135 \]
Subtract \( 53 \) from both sides:
\[ 41x = 135 - 53 \]
\[ 41x = 82 \]

### Step 5: Solve for \( x \)
Now, divide both sides by \( 41 \):
\[ x = \frac{82}{41} = 2 \]

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