What are the values of x and y in the system of linear equations `4x + 5y = -3` and `2x - y = 2` ?

Options

A. (-3, -2)
B. (3, 2)
C. (-3, 2)
D. (3, -2)

Answer

To solve the system of linear equations given by:

1. \( 4x + 5y = -3 \) (Equation 1)
2. \( 2x - y = 2 \) (Equation 2)

we can use the method of elimination or substitution. Here, we'll use elimination.

### Step 1: Make the coefficients of \( y \) the same
To eliminate \( y \), we can multiply Equation 2 by 5 to match the coefficient of \( y \) in Equation 1.

\[ 5(2x - y) = 5(2) \]

This gives us:

\[ 10x - 5y = 10 \quad \text{(Equation 3)} \]

### Step 2: Rewrite the equations
Now we have:

1. \( 4x + 5y = -3 \) (Equation 1)
2. \( 10x - 5y = 10 \) (Equation 3)

### Step 3: Add the equations
Now, we can add Equation 1 and Equation 3 together to eliminate \( y \):

\[ (4x + 5y) + (10x - 5y) = -3 + 10 \]

This simplifies to:

\[ 14x = 7 \]

### Step 4: Solve for \( x \)
Now, divide both sides by 14:

\[ x = \frac{7}{14} = \frac{1}{2} \]

### Step 5: Substitute \( x \) back into one of the original equations
Now that we have \( x \), we can substitute it back into Equation 2 to find \( y \):

\[ 2x - y = 2 \]

Substituting \( x = \frac{1}{2} \):

\[ 2\left(\frac{1}{2}\right) - y = 2 \]

This simplifies to:

\[ 1 - y = 2 \]

### Step 6: Solve for \( y \)
Now, isolate \( y \):

\[ -y = 2 - 1 \]
\[ -y = 1 \]
\[ y = -1 \]

### Final Values
Thus, the values of \( x \) and \( y \) are:

\[ x = \frac{1}{2}, \quad y = -1 \]

### Summary
The solution to the system of equations is:
- \( x = \frac{1}{2} \)
- \( y = -1 \)

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