`{(-2x+5y=1), (7x-10y=-11):}
If (x,y) is a solution to the system of equations above, what is the sum of x and y?

Options

A. `-(137)/(30)`
B. `-4`
C. `-(10)/(3)`
D. `-3`

Answer

To solve the system of equations given by:

1. \(-2x + 5y = 1\) (Equation 1)
2. \(7x - 10y = -11\) (Equation 2)

we will use the elimination method. Here are the steps:

### Step 1: Multiply the first equation
We can multiply the first equation by 2 to make the coefficients of \(y\) in both equations compatible for elimination.

\[ 2(-2x + 5y) = 2(1) \implies -4x + 10y = 2 \quad \text{(Equation 3)} \]

### Step 2: Write the second equation
The second equation remains unchanged:

\[ 7x - 10y = -11 \quad \text{(Equation 2)} \]

### Step 3: Add the two equations
Now, we will add Equation 3 and Equation 2:

\[ (-4x + 10y) + (7x - 10y) = 2 + (-11) \]

This simplifies to:

\[ (-4x + 7x) + (10y - 10y) = 2 - 11 \]

\[ 3x + 0 = -9 \]

### Step 4: Solve for \(x\)
Now, we can solve for \(x\):

\[ 3x = -9 \implies x = \frac{-9}{3} = -3 \]

### Step 5: Substitute \(x\) back to find \(y\)
Now that we have \(x = -3\), we can substitute this value back into Equation 1 to find \(y\):

\[ -2(-3) + 5y = 1 \]

This simplifies to:

\[ 6 + 5y = 1 \]

Subtract 6 from both sides:

\[ 5y = 1 - 6 \implies 5y = -5 \]

Now, divide by 5:

\[ y = \frac{-5}{5} = -1 \]

### Step 6: Find the sum of \(x\) and \(y\)
Now that we have both \(x\) and \(y\):

\[ x + y = -3 + (-1) = -3 - 1 = -4 \]

Thus, the sum of \(x\) and \(y\) is \(-4\).

### Final Answer
The sum of \(x\) and \(y\) is \(-4\).

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