Solving 2 variables linear equations by cross multiplication method
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Answer:
In practice, the method of cross-multiplying means that we multiply the numerator of each (or one) side by the denominator of the other side, effectively crossing the terms over. Each step in these procedures is based on a single, fundamental property of equations.
Step-by-step explanation:
In general, a pair of linear equations in two variables can be represented as a1 x + b1 y + c1 = 0 and a2 x + b2 y + c2 = 0.
For solving pair of linear equations in two variables following steps are followed:
Given a pair of linear equations in two variables;
a1 x + b1 y + c1 = 0 —(1)
a2 x + b2 y + c2 = 0 —(2)
Step i) Equation (1) is multiplied with b2 and Equation (2) by b1.
b2 a1 x + b2 b1 y + b2 c1 = 0 —(3)
b1 a2 x + b1 b2 y + b1 c2 = 0 —(4)
Step ii) Subtracting Equation (4) from (3):
(b2 a1 − b1 a2)x + (b2 b1 − b1 b2)y + (b2 c1 − b1 c2) = 0
⇒(b2 a1 − b1 a2)x = b1 c2 − b2 c1
⇒ x = b1 c2 − b2 c1b2 a1 − b1 a2 , given b2 a1 − b1 a2 ≠ 0
Step iii) The value of x obtained as such is substituted either in Equation (1) or Equation (2).Hence, the value of y obtained is:
y = c1 a2 − c2 a1b2 a1 − b1 a2
The solution of the equations is given as:
xb1 c2 − b2 c1 = yc1 a2 − c2 a1 = 1b2 a1 − b1 a2 —(5)
The above method is called is called ‘Cross-Multiplication Method’, as following cross-multiplication technique can be used to simplify the solution and hence will help in memorizing it. To memorize the method of cross multiplication, for solving linear equation in two variables the following diagram is helpful:
Cross Multiplication Method
The arrows indicate the multiplication of the values connected through the arrow and then the second product is subtracted from the first. The obtained result is then substituted as the denominator of the variables and 1, as mentioned above the arrow and then the entire values obtained are equated to form equation (5).
xb1 c2 − b2 c1 = yc1 a2 − c2 a1 = 1b2 a1 − b1 a2
From here, x and y are evaluated provided that a1 b2 − a2 b1 ≠ 0. Therefore, this method is called as cross multiplication as depicted.
In such a method, the condition for consistency of pair of linear equation in two variables must be checked which are as follows:
If a1a2 ≠ b1b2 , then we get a unique solution and the pair of linear equations in two variables are consistent.
If a1a2 = b1b2 = c1c2 , then there exists infinitely many solutions and the pair of lines are coincident and therefore, dependent and consistent.
If a1a2 = b1b2 ≠ c1c2 , then there exists no solution and the pair of linear equations in two variables are said to be inconsistent.