` 3 x - 5y - 19 = 0 `,
` - 7x + 3y + 1 = 0`

Options

A. ` x =2 , y = - 5`.
B. ` x =-2 , y = 5`.
C. ` x =-12 , y = - 5`.
D. ` x =-2 , y = - 15`.

Answer

To solve the system of equations:

1. \( 3x - 5y - 19 = 0 \)
2. \( -7x + 3y + 1 = 0 \)

we will follow these steps:

### Step 1: Rewrite the equations in standard form
We can rewrite the equations by isolating the constants on one side.

From the first equation:
\[ 3x - 5y = 19 \quad \text{(Equation 1)} \]

From the second equation:
\[ -7x + 3y = -1 \quad \text{(Equation 2)} \]

### Step 2: Make the coefficients of \( y \) equal
To eliminate \( y \), we can multiply the first equation by 3 and the second equation by 5 to make the coefficients of \( y \) equal.

Multiplying Equation 1 by 3:
\[ 3(3x - 5y) = 3(19) \implies 9x - 15y = 57 \quad \text{(Equation 3)} \]

Multiplying Equation 2 by 5:
\[ 5(-7x + 3y) = 5(-1) \implies -35x + 15y = -5 \quad \text{(Equation 4)} \]

### Step 3: Add the two new equations
Now we will add Equation 3 and Equation 4 to eliminate \( y \):
\[ (9x - 15y) + (-35x + 15y) = 57 - 5 \]
This simplifies to:
\[ 9x - 35x = 52 \implies -26x = 52 \]

### Step 4: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{52}{-26} = -2 \]

### Step 5: Substitute \( x \) back into one of the original equations
We can substitute \( x = -2 \) back into Equation 1 to find \( y \):
\[ 3(-2) - 5y = 19 \]
This simplifies to:
\[ -6 - 5y = 19 \]
Adding 6 to both sides gives:
\[ -5y = 25 \]
Now, divide by -5:
\[ y = \frac{25}{-5} = -5 \]

### Final Solution
Thus, the solution to the system of equations is:
\[ x = -2, \quad y = -5 \]