Solve the following simultaneous equations :
` 4x + 3y = 4 , 6x + 5y = 8 `

Answer

To solve the simultaneous equations \( 4x + 3y = 4 \) and \( 6x + 5y = 8 \), we will use the elimination method. Here are the steps:

### Step 1: Write down the equations
We have the following two equations:
1. \( 4x + 3y = 4 \) (Equation 1)
2. \( 6x + 5y = 8 \) (Equation 2)

### Step 2: Multiply the equations to align coefficients
To eliminate one variable, we will multiply the first equation by 6 and the second equation by 4. This will help us align the coefficients of \( x \).

- Multiply Equation 1 by 6:
\[ 6(4x + 3y) = 6(4) \implies 24x + 18y = 24 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by 4:
\[ 4(6x + 5y) = 4(8) \implies 24x + 20y = 32 \quad \text{(Equation 4)} \]

### Step 3: Subtract the equations
Now, we will subtract Equation 4 from Equation 3 to eliminate \( x \):
\[ (24x + 18y) - (24x + 20y) = 24 - 32 \]
This simplifies to:
\[ 18y - 20y = -8 \implies -2y = -8 \]

### Step 4: Solve for \( y \)
Now, divide both sides by -2:
\[ y = \frac{-8}{-2} = 4 \]

### Step 5: Substitute \( y \) back into one of the original equations
Now that we have \( y = 4 \), we will substitute this value back into Equation 1 to find \( x \):
\[ 4x + 3(4) = 4 \]
This simplifies to:
\[ 4x + 12 = 4 \]
Subtract 12 from both sides:
\[ 4x = 4 - 12 \implies 4x = -8 \]

### Step 6: Solve for \( x \)
Now, divide both sides by 4:
\[ x = \frac{-8}{4} = -2 \]

### Final Solution
Thus, the solution to the simultaneous equations is:
\[ (x, y) = (-2, 4) \]