The cross multiplication method is a mathematical technique used to solve equations involving two fractions or ratios.
Solved Examples
Example 1: Solve the following linear equations using the cross-multiplication method.
- 2x - 3y = 6
- 5x - 6y = 8
Given equations are,
2x - 3y = 6 and 5x - 6y = 8
Comparing with a1x + b1y = -c1 and a2x + b2y = -c2
a1 = 2, b1 = -3, c1 = -6
a2 = 5, b2 = -6, c2 = -8
Using Cross Multiplication Method,
x/(b1c2 - b2c1) = y/(c1a2 - c2a1) = 1/(b2a1 - b1a2)
Substituting all the values
x/[(-3).(-8) - (-6).(-6)] = y/[(-6).(5) - (-8).(2)] = 1/[(-6).(2) - (-3).(5)]
x/[24 - 36] = y/[-30 + 16] = 1/[-12 + 15]
x/(-12) = y/(-14) = 1/(3)
x/(-12) = 1/3
x = -4
y/(-14) = 1/(3)
y = -14/3
Thus, the value of x = -4 and -14/3
Example 2: Solve the following linear equations using the cross-multiplication method.
- 3x + 4y = 5
- x + 2y = 6
Given equations are,
3x + 4y = 5 and x + 2y = 6
Comparing with a1x + b1y = -c1 and a2x + b2y = -c2
a1 = 3, b1 = 4, c1 = -5
a2 = 1, b2 = 2, c2 = -6
Using Cross Multiplication Method,
x/(b1c2 - b2c1) = y/(c1a2 - c2a1) = 1/(b2a1 - b1a2)
Substituting all the values
x/[(4).(-6) - (2).(-5)] = y/[(-5).(1) - (-6).(3)] = 1/[(2).(3) - (4).(1)]
x/[-24 + 10] = y/[-5 + 18] = 1/[6 - 4]
x/(-14) = y/(13) = 1/(2)
x/(-14) = 1/2
x = -7
y/(13) = 1/(2)
y = 13/2
Thus, the value of x = -7 and 13/2
Example 3: Check whether 2x + 3y = 5 and 3x + y = 7 yields a unique solution or not.
Given Equation,
- 2x + 3y = 5
- 3x + y = 7
Comparing with a1x + b1y = -c1 and a2x + b2y = -c2
a1 = 2, b1 = 3
a2 = 3, b2 = 1
Using the condition
a1/a2 ≠ b1/b2
2/3 ≠ 3/1
As the condition holds true the given equations give the unique solution.
Practice Problems
Problem 1: Solve the pair of equations: 2x+y = 7 and 3x−y = 8
Problem 2: Find the values of x and y: 4x+5y=18 and 2x−y=4.
Problem 3: Solve the following system of equations: 3x+2y = 12 and 5x−y = 7
Problem 4: Using the Cross Multiplication Method, find the solution of: 7x + 3y = 20 and 2x + y = 5
Problem 5: Determine the values of x and y: 5x−2y = 16 and 3x+4y = 18
