The solution of the equations:
`3(x+3y-1)=2(y-3x+1)` and `3(5y-7x-3)=2(5x+7y+12)` are

Options

A. `x=1,y=2`
B. `x=-1,y=2`
C. `x=1,y=-2`
D. `x=-1,y=-2`

Answer

To solve the equations \(3(x + 3y - 1) = 2(y - 3x + 1)\) and \(3(5y - 7x - 3) = 2(5x + 7y + 12)\), we will follow these steps:

### Step 1: Simplify the first equation
Start with the first equation:
\[ 3(x + 3y - 1) = 2(y - 3x + 1) \]
Distributing both sides:
\[ 3x + 9y - 3 = 2y - 6x + 2 \]
Rearranging terms gives:
\[ 3x + 9y - 2y + 6x - 3 - 2 = 0 \]
This simplifies to:
\[ 9x + 7y - 5 = 0 \quad \text{(Equation 1)} \]

### Step 2: Simplify the second equation
Now simplify the second equation:
\[ 3(5y - 7x - 3) = 2(5x + 7y + 12) \]
Distributing both sides:
\[ 15y - 21x - 9 = 10x + 14y + 24 \]
Rearranging terms gives:
\[ 15y - 14y - 21x - 10x - 9 - 24 = 0 \]
This simplifies to:
\[ -31x + y - 33 = 0 \quad \text{(Equation 2)} \]

### Step 3: Multiply Equation 2
To eliminate \(y\), we can multiply Equation 2 by 7:
\[ 7(-31x + y - 33) = 0 \]
This gives:
\[ -217x + 7y - 231 = 0 \quad \text{(Equation 3)} \]

### Step 4: Add Equation 1 and Equation 3
Now add Equation 1 and Equation 3 to eliminate \(y\):
\[ (9x + 7y - 5) + (-217x + 7y - 231) = 0 \]
This simplifies to:
\[ (9x - 217x) + (7y + 7y) - (5 + 231) = 0 \]
Which gives:
\[ -208x + 14y - 236 = 0 \]

### Step 5: Solve for \(x\)
From the equation above, we can isolate \(x\):
\[ -208x = 236 - 14y \]
Now we can express \(x\) in terms of \(y\):
\[ x = \frac{236 - 14y}{208} \]

### Step 6: Substitute \(x\) back into Equation 1
Now substitute \(x\) back into Equation 1 to find \(y\):
\[ 9\left(\frac{236 - 14y}{208}\right) + 7y - 5 = 0 \]
Multiply through by 208 to eliminate the fraction:
\[ 9(236 - 14y) + 7(208y) - 1040 = 0 \]
This simplifies to:
\[ 2124 - 126y + 1456y - 1040 = 0 \]
Combining like terms gives:
\[ 1330y + 1084 = 0 \]
Solving for \(y\):
\[ y = \frac{-1084}{1330} = 2 \]

### Step 7: Substitute \(y\) back to find \(x\)
Now substitute \(y = 2\) back into Equation 1:
\[ 9x + 7(2) - 5 = 0 \]
This simplifies to:
\[ 9x + 14 - 5 = 0 \]
So:
\[ 9x + 9 = 0 \implies x = -1 \]

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