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.
4m + 6n = 54    ...(i)
3m + 2n = 28    ...(ii)

Step 2: Write the determinant of coefficients (Δ).
Δ = | 4 6 |
     | 3 2 |
= (4 × 2) − (6 × 3) = 8 − 18 = −10

Step 3: Find determinant Δ_m (replace first column by constants).
Δ_m = | 54 6 |
     | 28 2 |
= (54 × 2) − (6 × 28) = 108 − 168 = −60

Step 4: Find determinant Δ_n (replace second column by constants).
Δ_n = | 4 54 |
     | 3 28 |
= (4 × 28) − (54 × 3) = 112 − 162 = −50

Step 5: Apply Cramer’s rule.
m = Δ_m / Δ = (−60) / (−10) = 6
n = Δ_n / Δ = (−50) / (−10) = 5

Step 6: Conclusion.
Hence, the solution of the given simultaneous equations is:
m = 6,   n = 5

Final Answer:
m = 6,   n = 5