Question

How do you solve the systems of equations : $x + 2y = 6$; $x - 3y = - 4$?

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 terms containing x or y to have the same coefficient 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 unknown variable we substitute it back in any one of the equations and get the value of another unknown variable.

Complete step-by-step answer:
Given two linear equations,
$x + 2y = 6$ …… (1)
$x - 3y = - 4$ …… (2)
We are asked to solve the system of equations given above.
If we carefully observe the two equations, the coefficient of the variable x is the same which is equal to one. So, there is no need to multiply them with a suitable number to eliminate the terms of the same coefficient using addition or subtraction.
Now we subtract equation (2) from (1), we get,
$ \Rightarrow x + 2y - (x - 3y) = 6 - ( - 4)$
Simplifying this we get,
$ \Rightarrow x + 2y - x + 3y = 6 + 4$
Rearranging the terms we get,
$ \Rightarrow x - x + 2y + 3y = 6 + 4$
Combining the like terms $x - x = 0$ and $2y + 3y = 5y$
Hence the above equation becomes,
$ \Rightarrow 0 + 5y = 10$
$ \Rightarrow 5y = 10$
Taking 5 to the right hand side we get,
$ \Rightarrow y = \dfrac{{10}}{5}$
$ \Rightarrow y = 2$.
Now to get the value of x we substitute back $y = 2$ in equation (1) or (2).
Substituting $y = 2$ in the equation (1), we get,
 $ \Rightarrow x + 2(2) = 6$
$ \Rightarrow x + 4 = 6$
Taking 4 to the right hand side we get,
$ \Rightarrow x = 6 - 4$
$ \Rightarrow x = 2$
Hence the values of unknown variables for the system of equations $x + 2y = 6$ and $x - 3y = - 4$ are given by $x = 2$ and $y = 2$.

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 coefficient.
We can verify whether the obtained values of the variables x and y are correct, by substituting them 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 it 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.