Solve the equations `4x+3y = 9 and x+y = 2` for x and y.

Options

A. `x=3, y=-1`
B. `x=1, y=-3`
C. `x=3, y=2`
D. `x=-1, y=3`

Answer

To solve the equations \(4x + 3y = 9\) and \(x + y = 2\) for \(x\) and \(y\), we can use the substitution or elimination method. Here, we'll use the elimination method.

### Step 1: Write down the equations
We have:
1. \(4x + 3y = 9\) (Equation 1)
2. \(x + y = 2\) (Equation 2)

### Step 2: Make the coefficients of \(x\) equal
To eliminate \(x\), we can multiply Equation 2 by 4, so that the coefficient of \(x\) in both equations becomes the same.

Multiply Equation 2 by 4:
\[ 4(x + y) = 4(2) \]
This gives us:
\[ 4x + 4y = 8 \quad \text{(Equation 3)} \]

### Step 3: Subtract Equation 1 from Equation 3
Now we will subtract Equation 1 from Equation 3:
\[ (4x + 4y) - (4x + 3y) = 8 - 9 \]
This simplifies to:
\[ 4y - 3y = -1 \]
So:
\[ y = -1 \]

### Step 4: Substitute \(y\) back into one of the original equations
Now that we have \(y = -1\), we can substitute this value back into Equation 2 to find \(x\):
\[ x + (-1) = 2 \]
This simplifies to:
\[ x - 1 = 2 \]
So:
\[ x = 2 + 1 = 3 \]

### Step 5: Write the solution
The solution to the equations is:
\[ x = 3, \quad y = -1 \]

### Summary of the solution
The values of \(x\) and \(y\) that satisfy both equations are:
\[ \boxed{(3, -1)} \]