Question

Solve the following pair of equations:
\[ 3x - 5y - 4 = 0 \] \[ 9x = 2y + 7 \]

Hint:

To solve simultaneous linear equations, use elimination or substitution method. Make coefficients of one variable equal and subtract to eliminate it.

Answer 1

Step 1: Write both equations in standard form.
\[ 3x - 5y - 4 = 0 \Rightarrow 3x - 5y = 4 \quad \text{(i)} \] \[ 9x = 2y + 7 \Rightarrow 9x - 2y = 7 \quad \text{(ii)} \]
Step 2: Eliminate one variable.
We will eliminate \( y \). Multiply equation (i) by 2 and equation (ii) by 5 to make coefficients of \( y \) equal. \[ 2(3x - 5y) = 2(4) \Rightarrow 6x - 10y = 8 \quad \text{(iii)} \] \[ 5(9x - 2y) = 5(7) \Rightarrow 45x - 10y = 35 \quad \text{(iv)} \]
Step 3: Subtract (iii) from (iv).
\[ (45x - 10y) - (6x - 10y) = 35 - 8 \] \[ 39x = 27 \] \[ x = \frac{27}{39} = \frac{9}{13} \]
Step 4: Substitute in equation (i).
\[ 3\left(\frac{9}{13}\right) - 5y = 4 \] \[ \frac{27}{13} - 5y = 4 \] \[ -5y = 4 - \frac{27}{13} = \frac{52 - 27}{13} = \frac{25}{13} \] \[ y = -\frac{5}{13} \] Step 5: Final Answer.
\[ \boxed{x = \frac{9}{13}, \, y = -\frac{5}{13}} \]