Solve `(x+2)/(y+2)+2=0,(x-4)/(y-2)=(x-1)/(y+7)`

Options

A. x=2,y=4
B. x=-2,y=4
C. x=2,y=-4
D. x=-2,y=-4

Answer

To solve the system of equations given by

\[ \frac{x+2}{y+2} + 2 = 0 \]
and
\[ \frac{x-4}{y-2} = \frac{x-1}{y+7}, \]

we will follow these steps:

### Step 1: Simplify the first equation

Start with the first equation:

\[ \frac{x+2}{y+2} + 2 = 0. \]

Subtract 2 from both sides:

\[ \frac{x+2}{y+2} = -2. \]

Now, cross-multiply:

\[ x + 2 = -2(y + 2). \]

Expanding the right side gives:

\[ x + 2 = -2y - 4. \]

Rearranging this equation leads to:

\[ x + 2y + 6 = 0. \]

This simplifies to:

\[ x + 2y = -6. \quad \text{(Equation 1)} \]

### Step 2: Simplify the second equation

Now, consider the second equation:

\[ \frac{x-4}{y-2} = \frac{x-1}{y+7}. \]

Cross-multiply:

\[ (x - 4)(y + 7) = (x - 1)(y - 2). \]

Expanding both sides gives:

\[ xy + 7x - 4y - 28 = xy - 2x + y - 2. \]

Now, cancel \(xy\) from both sides:

\[ 7x - 4y - 28 = -2x + y - 2. \]

Rearranging this leads to:

\[ 7x + 2x - 4y - y = 28 - 2. \]

This simplifies to:

\[ 9x - 5y = 26. \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations

Now we have the system of equations:

1. \(x + 2y = -6\)
2. \(9x - 5y = 26\)

From Equation 1, we can express \(x\) in terms of \(y\):

\[ x = -6 - 2y. \]

Substituting this expression for \(x\) into Equation 2:

\[ 9(-6 - 2y) - 5y = 26. \]

Expanding this gives:

\[ -54 - 18y - 5y = 26. \]

Combining like terms results in:

\[ -54 - 23y = 26. \]

Adding 54 to both sides:

\[ -23y = 80. \]

Dividing both sides by -23 gives:

\[ y = -\frac{80}{23}. \]

### Step 4: Substitute back to find \(x\)

Now substitute \(y\) back into the expression for \(x\):

\[ x = -6 - 2\left(-\frac{80}{23}\right). \]

Calculating this gives:

\[ x = -6 + \frac{160}{23} = -\frac{138}{23} + \frac{160}{23} = \frac{22}{23}. \]

### Final Solution

Thus, the solution to the system of equations is:

\[ x = \frac{22}{23}, \quad y = -\frac{80}{23}. \]