Question

Solve the following pair of linear equations by cross multiplication method-
$x + 2y = 2$
$x - 3y = 7$

Answer 1

- Hint: The cross multiplication method for two equations of the general form is given by-
$\begin{align}
  & {{\text{a}}_1}{\text{x}} + {{\text{b}}_1}{\text{y}} + {{\text{c}}_1} = 0 \\
  & {{\text{a}}_2}{\text{x}} + {{\text{b}}_2}{\text{y}} + {{\text{c}}_2} = 0 \\
\end{align} $
The solution for the equation is given by-
$
\dfrac{{\text{x}}}{{{{\text{b}}_1}{{\text{c}}_2} - {{\text{b}}_2}{{\text{c}}_1}}} = \dfrac{{\text{y}}}{{{{\text{a}}_2}{{\text{c}}_1} - {{\text{a}}_1}{{\text{c}}_2}}} = \dfrac{1}{{{{\text{a}}_1}{{\text{b}}_2} - {{\text{a}}_2}{{\text{b}}_1}}} \\
$

Complete step-by-step solution -
The given equations can are-
$x + 2y = 2$
$x + 2y - 2 = 0…(1)$
$x - 3y = 7$
$x - 3y - 7 = 0…(2)$
From equations (1) and (2), we can see that-
$\begin{align}
 & {{\text{a}}_1} = 1,\;{{\text{b}}_1} = 2,\;{{\text{c}}_1} = - 2 \\
 & {{\text{a}}_2} = 1,\;{{\text{b}}_2} = - 3,\;{{\text{c}}_2} = - 7 \\
\end{align} $


The equations can be written as-
$\begin{align}
 & \dfrac{{\text{x}}}{{2\left( { - 7} \right) - \left( { - 3} \right)\left( { - 2} \right)}} = \dfrac{{\text{y}}}{{1\left( { - 2} \right) - 1\left( { - 7} \right)}} = \dfrac{1}{{1\left( { - 3} \right) - 1\left( 2 \right)}}\dfrac{{\text{x}}}{{ - 14 - 6}} = \dfrac{{\text{y}}}{{ - 2 + 7}} = \dfrac{1}{{ - 3 - 2}} \\
  & \dfrac{{\text{x}}}{{ - 20}} = \dfrac{{\text{y}}}{5} = - \dfrac{1}{5} \\
 & \dfrac{{\text{x}}}{{ - 20}} = - \dfrac{1}{5}\;and\;\dfrac{{\text{y}}}{5} = - \dfrac{1}{5} \\
 & {\text{x}}\; = \;4\;and\;{\text{y}}\; = \; - 1 \\
\end{align} $

Note: Whenever we are asked to solve equations of any kind, we should always check the answer obtained by substituting the values of x and y in the equation and see if they satisfy the equations or not. This is the best method for verification of the answer. In this question-
$x + 2y - 2 = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x - 3y - 7 = 0$
$4 + 2(-1) - 2 = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,4 - 3(-1) - 7 = 0$
$4 - 2 - 2 = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;4 + 3 - 7 = 0$
$0 = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,0 = 0$
The solution is thus verified.