Question

Solve \( 2x + 3y = 11 \) and \( 2x - 4y = -24 \) and hence find the value of \( m \) for which \( y = mx + 3 \).

Hint:

When solving a system of linear equations, substitution or elimination methods can be used to find the values of the variables. In this case, substitution was used to solve for \( y \) and \( x \).

Answer 1

We are given the system of equations: \[ 2x + 3y = 11 \quad \text{(1)} \] \[ 2x - 4y = -24 \quad \text{(2)}. \]
Step 1: Solve equation (1) for \( x \): \[ 2x = 11 - 3y \quad \Rightarrow \quad x = \frac{11 - 3y}{2}. \]
Step 2: Substitute this expression for \( x \) into equation (2): \[ 2\left( \frac{11 - 3y}{2} \right) - 4y = -24. \] Simplify: \[ 11 - 3y - 4y = -24 \quad \Rightarrow \quad 11 - 7y = -24. \]
Step 3: Solve for \( y \): \[ -7y = -24 - 11 \quad \Rightarrow \quad -7y = -35 \quad \Rightarrow \quad y = 5. \]
Step 4: Substitute \( y = 5 \) into the equation for \( x \): \[ x = \frac{11 - 3(5)}{2} = \frac{11 - 15}{2} = \frac{-4}{2} = -2. \]
Step 5: Now, use the equation \( y = mx + 3 \) and substitute \( x = -2 \) and \( y = 5 \) to find \( m \): \[ 5 = m(-2) + 3. \] Solve for \( m \): \[ 5 = -2m + 3 \quad \Rightarrow \quad -2m = 5 - 3 = 2 \quad \Rightarrow \quad m = -1. \]
Conclusion: The value of \( m \) is \( -1 \).