Question

Solve the following simultaneous equations using Cramer’s Rule: \[ 4m + 6n = 54 \quad \text{and} \quad 3m + 2n = 28 \]

Hint:

Cramer’s Rule is a direct method for solving two linear equations using determinants: \[ x = \frac{\Delta_x}{\Delta}, \quad y = \frac{\Delta_y}{\Delta} \] Always compute \(\Delta \neq 0\) before applying the rule.

Answer 1

Step 1: Write the given equations in standard form.
\[ \begin{aligned} 4m + 6n &= 54 \quad \text{...(i)} \\ 3m + 2n &= 28 \quad \text{...(ii)} \end{aligned} \] Step 2: Write the determinant of coefficients (\(\Delta\)).
\[ \Delta = \begin{vmatrix} 4 & 6 \\ 3 & 2 \end{vmatrix} = (4)(2) - (6)(3) = 8 - 18 = -10 \] Step 3: Find determinant \(\Delta_m\) (replace first column by constants).
\[ \Delta_m = \begin{vmatrix} 54 & 6 \\ 28 & 2 \end{vmatrix} = (54)(2) - (6)(28) = 108 - 168 = -60 \] Step 4: Find determinant \(\Delta_n\) (replace second column by constants).
\[ \Delta_n = \begin{vmatrix} 4 & 54 \\ 3 & 28 \end{vmatrix} = (4)(28) - (54)(3) = 112 - 162 = -50 \] Step 5: Apply Cramer’s rule.
\[ m = \frac{\Delta_m}{\Delta} = \frac{-60}{-10} = 6 \] \[ n = \frac{\Delta_n}{\Delta} = \frac{-50}{-10} = 5 \] Step 6: Conclusion.
Hence, the solution of the given simultaneous equations is \(m = 6, \; n = 5.\)
Final Answer: \[ \boxed{m = 6, \; n = 5} \]