Solve 11(x - 5) + 10(y - 2) + 54 = 0
7(2x - 1) + 9 (3y - 1) = 25

Answer

To solve the simultaneous equations given by:

1. \( 11(x - 5) + 10(y - 2) + 54 = 0 \)
2. \( 7(2x - 1) + 9(3y - 1) = 25 \)

we will follow these steps:

### Step 1: Simplify the first equation

Starting with the first equation:

\[ 11(x - 5) + 10(y - 2) + 54 = 0 \]

Distributing the terms inside the parentheses:

\[ 11x - 55 + 10y - 20 + 54 = 0 \]

Combining like terms:

\[ 11x + 10y - 75 + 54 = 0 \]

This simplifies to:

\[ 11x + 10y - 21 = 0 \]

Rearranging gives us the first equation:

\[ 11x + 10y = 21 \quad \text{(Equation 1)} \]

### Step 2: Simplify the second equation

Now, let's simplify the second equation:

\[ 7(2x - 1) + 9(3y - 1) = 25 \]

Distributing the terms:

\[ 14x - 7 + 27y - 9 = 25 \]

Combining like terms:

\[ 14x + 27y - 16 = 25 \]

Rearranging gives us the second equation:

\[ 14x + 27y = 41 \quad \text{(Equation 2)} \]

### Step 3: Solve the equations simultaneously

We now have the following two equations:

1. \( 11x + 10y = 21 \) (Equation 1)
2. \( 14x + 27y = 41 \) (Equation 2)

To eliminate one variable, we can multiply Equation 1 by 14 and Equation 2 by 11:

\[ 14(11x + 10y) = 14(21) \implies 154x + 140y = 294 \quad \text{(Equation 3)} \]
\[ 11(14x + 27y) = 11(41) \implies 154x + 297y = 451 \quad \text{(Equation 4)} \]

### Step 4: Subtract the equations

Now, we will subtract Equation 3 from Equation 4:

\[ (154x + 297y) - (154x + 140y) = 451 - 294 \]

This simplifies to:

\[ 157y = 157 \]

Dividing both sides by 157 gives:

\[ y = 1 \]

### Step 5: Substitute \( y \) back into one of the original equations

Now that we have \( y = 1 \), we can substitute this value back into Equation 1:

\[ 11x + 10(1) = 21 \]

This simplifies to:

\[ 11x + 10 = 21 \]

Subtracting 10 from both sides:

\[ 11x = 11 \]

Dividing both sides by 11 gives:

\[ x = 1 \]

### Final Solution

Thus, the solution to the simultaneous equations is:

\[ x = 1, \quad y = 1 \]