Question

How do you solve $3x + y = 4$ and $2x + y = 1$ ?

Answer 1

Hint: Here we are given two linear equations in two variables x and y. We need to use elimination methods to find the value of unknowns x and y. Firstly, we will try to make the unknowns of x and 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 value. Then we add or subtract them to obtain the value of one unknown. If we get one unknown we substitute it back in any one of the equations and get the value of the other unknown.

Complete step by step solution:
Given two linear equations,
$3x + y = 4$ …… (1)
$2x + y = 1$ …... (2)
If we carefully observe the above equations, the coefficient of the variable y is the same. i.e. the coefficient of y is 1.
So we subtract the equation (2) from (1), we get,
$ \Rightarrow 3x + y - 2x - y = 4 - 1$
Rearranging the terms we get,
$ \Rightarrow 3x - 2x + y - y = 4 - 1$
Combining the like terms $3x - 2x = 1$ and $y - y = 0$
Hence the above equation becomes,
$ \Rightarrow x + 0 = 3$
$ \Rightarrow x = 3$.
To obtain the value of the one more unknown y, we substitute the value of x in any one of the above equations
i.e. to get the value of y we substitute back $x = 3$ in the equation (1) or (2).
Substituting $x = 3$ in the equation (1), we get,
$ \Rightarrow 3(3) + y = 4$
$ \Rightarrow 9 + y = 4$
Taking 9 to the other side we get,
$ \Rightarrow y = 4 - 9$
$ \Rightarrow y = - 5$

Hence the values of unknown are given by $x = 3$ and $y = - 5$.

Note: Students 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 values 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.