Question
Solve the following pair of linear equations using the cross multiplication method:
Solution — Step by Step
Make sure both equations are in the form a₁x + b₁y + c₁ = 0. Rearrange:
So: a₁ = 2, b₁ = 3, c₁ = -8 and a₂ = 4, b₂ = -1, c₂ = -2.
Write the coefficients in this arrangement — and here’s the WHY: the formula works by eliminating one variable at a time through “cross multiplying” the coefficients diagonally.
A visual trick that thousands of students use: write the columns b, c, a, b and cross-multiply adjacent pairs.
Plug in the values:
We get:
From the first and third ratios: x/(-14) = 1/(-14), so x = 1.
From the second and third ratios: y/(-28) = 1/(-14), so y = (-28)/(-14) = y = 2.
Always verify — this takes 10 seconds and saves marks in board exams.
Equation 1: 2(1) + 3(2) = 2 + 6 = 8 ✓
Equation 2: 4(1) - 2 = 4 - 2 = 2 ✓
Answer: x = 1, y = 2
Why This Works
The cross multiplication method is essentially Cramer’s Rule dressed up for Class 10. Each “determinant” you calculate represents the area of a parallelogram formed by two coefficient vectors — when this area is non-zero, the system has a unique solution.
The denominator a₁b₂ - a₂b₁ is the key quantity. If it equals zero, the lines are either parallel (no solution) or the same line (infinite solutions). This is exactly why we check this value first in more advanced problems.
The formula looks intimidating at first glance, but once you internalize the pattern — b, c coefficients for x; c, a coefficients for y; a, b coefficients for the denominator — it becomes mechanical. CBSE rewards this method in their marking scheme because each determinant is a separate step worth marks.
Alternative Method
We can solve the same system using substitution to cross-check.
From equation 2: 4x - y = 2, so y = 4x - 2.
Substitute into equation 1:
Then y = 4(1) - 2 = 2.
Same answer. Cross multiplication is faster when both equations are “messy” with no obvious variable to isolate. Substitution is faster when one equation gives you a clean expression like y = ....
In CBSE 2024 Board Exam, cross multiplication questions always have “neat” integer answers. If you’re getting fractions in a 3-mark question, recheck your signs — the most common error is a sign flip in the c terms when rearranging to standard form.
Common Mistake
Sign error when moving constants to the left side. Students write 2x + 3y = 8 as a₁ = 2, b₁ = 3, c₁ = 8 (positive 8), forgetting that standard form is ax + by + c = 0, so c₁ = -8. This flips the sign of every determinant involving c, and your final answer will be wrong even though your method is perfectly applied. Always move everything to the left side first, then read off the coefficients.