Cross Multiplication Method in Linear Equation at Algebra Den

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Home >> Polynomials >> Linear Equations >> Solving a Pair of Linear Equations >> Cross Multiplication Method >>

Cross Multiplication Method in Linear Equation

Cross Multiplication Method Elimination Method Substitution Method

Under this method, we use the following formula to find solution to pair of linear equation:

x / b₁ c₂ - b₂ c₁ = y / c₁ a₂ - c₂ a₁ = 1 / a₁ b₂ - a₂ b₁

In order to remember the above formula, you may refer to the following diagram

In the above diagram, arrow symbolizes that two number are to be multiplied and 2nd product is to be subtracted from the 1st product.

Note: This cross multiplication applies to the situation only where a1b2 - a2b1 ≠ 0

Lets understand this method with the help of following example:

Example 1: Solve the following pair of examples:
2x + y = 10
5x + 2y = 23

Solution: Rearrange the given equations as shown below:
2x + y - 10 = 0 ... (equation 1)
5x + 2y - 23= 0 ... (equation 2)

Now let's label the value of variables as shown below:

a₁ = 2
a₂ = 5
b₁ = 1
b₂ = 2
c₁ = -10
c₂ = -23

Now, arrange the above variables as shown in above figure 1 and we get:

Apply the cross multiplication formula and we get;

x / (1 X -23) - (2 X-10) = y / (-10 X 5) - (-23 X 2) = 1 / (2 X 2) - (5 X 1)

solve brackets and we get:
x / -23 - (-20) = y / -50 - (-46) = 1 / 4 - 5
x / -23 + 20 = y / -50 + 46= 1 / 4 - 5
x / -3 = y / -4 = 1 / - 1
x / -3 = y / -4 = -1 / 1

or x / -3 = -1 / 1 and y / -4 = -1 / 1
Solving this we get,
x = 3 and y = 4

Hence, the solution is x = 3 and y = 4