`3x-4y=10`, `4x+3y=5`

Answer

To solve the system of equations given by \(3x - 4y = 10\) and \(4x + 3y = 5\), we will use the elimination method. Here are the steps to find the values of \(x\) and \(y\):

### Step 1: Write down the equations
We have the following two equations:
1. \(3x - 4y = 10\) (Equation 1)
2. \(4x + 3y = 5\) (Equation 2)

### Step 2: Multiply the equations to align coefficients
To eliminate \(y\), we will multiply Equation 1 by 3 and Equation 2 by 4. This will help us make the coefficients of \(y\) the same (but opposite in sign).

- Multiply Equation 1 by 3:
\[ 3(3x - 4y) = 3(10) \implies 9x - 12y = 30 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by 4:
\[ 4(4x + 3y) = 4(5) \implies 16x + 12y = 20 \quad \text{(Equation 4)} \]

### Step 3: Add the two new equations
Now we will add Equation 3 and Equation 4 together:
\[ (9x - 12y) + (16x + 12y) = 30 + 20 \]
This simplifies to:
\[ 25x = 50 \]

### Step 4: Solve for \(x\)
Now, we can solve for \(x\):
\[ x = \frac{50}{25} = 2 \]

### Step 5: Substitute \(x\) back into one of the original equations
Now that we have \(x = 2\), we will substitute this value back into Equation 1 to find \(y\):
\[ 3(2) - 4y = 10 \]
This simplifies to:
\[ 6 - 4y = 10 \]

### Step 6: Solve for \(y\)
Now, isolate \(y\):
\[ -4y = 10 - 6 \implies -4y = 4 \implies y = \frac{4}{-4} = -1 \]

### Final Solution
Thus, the solution to the system of equations is:
\[ x = 2, \quad y = -1 \]