Question

How do you solve the system of linear equations $x + 2y = 7$ and $2x - 3y = - 5$?

Answer 1

Hint: Here we are given two linear equations in two variables x and y. We will use elimination method to find the value of unknown variables x and y. Firstly, we will try to make the coefficient of x or y to have the same values in both the given equation. For this we multiply by a suitable number to both the equations and make the terms of x or y to contain the same coefficient. Then we add or subtract them to obtain the value of one unknown variable. If we get the value of one variable, we substitute it back in any one of the equations and get the value of another unknown variable.

Complete step-by-step solution:
Given two linear equations,
$x + 2y = 7$ …… (1)
$2x - 3y = - 5$ …… (2)
If we carefully observe the equations, there are no terms of x or y containing the same coefficient.
So, to make the unknown variables x and y to have the same coefficient, we multiply the equations with a suitable number and then we try to eliminate the terms with the same coefficient using addition or subtraction.
In this problem, we multiply the equation (1) by 2, so that the x terms will have the same coefficient in both the equations.
Multiplying the equation (1) by 2, we get,
$ \Rightarrow 2(x + 2y) = 2(7)$
$ \Rightarrow 2x + 4y = 14$ …… (3)
Now subtracting the equation (2) from (3), we get,
$ \Rightarrow 2x + 4y - (2x - 3y) = 14 - ( - 5)$
$ \Rightarrow 2x + 4y - 2x + 3y = 14 + 5$
Rearranging the terms we get,
$ \Rightarrow 2x - 2x + 4y + 3y = 14 + 5$
Combining the like terms $2x - 2x = 0$ and $4y = 3y = 7y$
Hence the above equation becomes,
$ \Rightarrow 0 + 7y = 19$
$ \Rightarrow 7y = 19$
Taking 7 to the right hand side we get,
$ \Rightarrow y = \dfrac{{19}}{7}$
Now to get the value of x we substitute back $y = \dfrac{{19}}{7}$ in equation (1) or (2).
Substituting $y = \dfrac{{19}}{7}$ in the equation (1), we get,
 $ \Rightarrow x + 2\left( {\dfrac{{19}}{7}} \right) = 7$
$ \Rightarrow x + \dfrac{{38}}{7} = 7$
Taking $\dfrac{{38}}{7}$ to the right hand side we get,
$ \Rightarrow x = 7 - \dfrac{{38}}{7}$
Taking LCM on the right hand side, we get,
$ \Rightarrow x = \dfrac{{49 - 38}}{7}$
$ \Rightarrow x = \dfrac{{11}}{7}$
Hence the solution for the equations $x + 2y = 7$ and $2x - 3y = - 5$ is given by $x = \dfrac{{11}}{7}$ and $y = \dfrac{{19}}{7}$.

Note: We must choose a suitable number to multiply the given linear equations to eliminate any one of the variables by making them to have the same value.
We can verify whether the obtained value of the variable x and y are correct, by substituting them with any one of the equations given. If the equation satisfies, then they are the required values.
We need to be careful while taking the terms to the other side. When transferring any variable or number to the other side, the sign of the same will be changed to its opposite sign.
It is important to know the following basic facts.
An equation remains unchanged or undisturbed if its satisfies the following conditions.
(1) If L.H.S. and R.H.S. are interchanged.
(2) If the same number is added on both sides of the equation.
(3) If the same number is subtracted on both sides of the equation.
(4) When both L.H.S. and R.H.S. are multiplied by the same number.
(5) When both L.H.S. and R.H.S. are divided by the same number.