Question

Solve the following pair of linear equations by cross multiplication method $2x+y=5,3x+2y=8$ .

Answer 1

- Hint: First look at the definition of Cross multiplication method carefully. Find the variables a, b, c, d, e, f from the equations given in the question. You must find variables by comparing given equations with $ax+by+c=0$ ; $dx+ey+f=0$. Now, use the relation between x, y and these assumed variables a, b, c, d, e, f and find the values of x, y. By formula equation you can get the variables x, y in terms of the newly assumed variables. Cross multiplication method equation:
$\dfrac{x}{bf-ec}=\dfrac{y}{cd-fa}=\dfrac{1}{ae-db}$

Complete step-by-step solution -

Cross Multiplication Method: In mathematics, cross multiplication method means we multiply the numerator of each side by the denominator of the other side, effectively crossing the terms over.
We write x, y, a, b, c, d, e, f as follows to get the formula: - Use cross multiplying after this to get x, y.
 

$\dfrac{x}{bf-ec}=\dfrac{y}{cd-fa}=\dfrac{1}{ae-db}$
By cross multiplication at first and last term, we get value of x:
$x=\dfrac{bf-ec}{ae-db}$ ………………………………………..(i)
By cross multiplying at second and last term, we get value of y:
$y=\dfrac{cd-fa}{ae-db}$ ………………………………………(ii)
Given equations in the question, are written in the form of:
 $2x+y=5,3x+2y=8$
By comparing these equations with $ax+by+c=0$ , $dx+ey+f=0$ we get variables as:
$a=2,b=1,c=-5,d=3,e=2,f=-8$
By substituting these values in equation (i), we get x as:
$x=\dfrac{\left( -8 \right)-\left( -10 \right)}{\left( 4 \right)-\left( 3 \right)}$
By simplifying, we get value of x as a constant: $x=2$
By substituting a, b, c, d, e, f in equation (ii), we get y as:
$y=\dfrac{\left( -15 \right)-\left( -16 \right)}{\left( 4 \right)-\left( 3 \right)}$
By simplifying the above equation, we get value of y as: \[y=1\]
Therefore, \[\left( 2,1 \right)\] is the solution of given equations.

Note: In cross multiplication method the representation is very crucial. Generally, students confuse and write a in place of b and vice-versa which will lead to wrong answers. Be careful while comparing you must make the right hand side zero. Generally, students take the constant as c but you must make sure the constant must be on the left hand side or else you will miss a negative sign and lead to the wrong answer.