Question

How do you solve $4x + 2y = 12$ and $x + 2y = 6$?

Answer 1

Hint: Here the given equations above are linear equations; therefore we will solve the equations by the method of elimination by subtracting the equations to reduce the variables in it. Finally we get the required answer.

Complete step-by-step solution:
We have the given equations as:
Equation $(1)$ can be written as: $4x + 2y = 12 \to (1)$
Equation $(2)$ can be written as: $x + 2y = 6 \to (2)$
Now since the equations are linear and the coefficient of $y$ is the same for both the equations, we don’t have to multiply or divide the equations, we will simplify both the equations just by simply subtracting them.
On subtracting equation $(2)$ from equation $(1)$ we get:
$ \Rightarrow 3x = 6$
On rearranging we get:
$ \Rightarrow x = \dfrac{6}{3}$
On simplifying we get:
$x = 2$, which is the value of $x$.
Now to get the value of $y$ we will substitute the value of $x = 3$ in equation $(1)$ .
On substituting we get:
 $ \Rightarrow 4(2) + 2y = 12$
On simplifying we get:
$ \Rightarrow 8 + 2y = 12$
On sending $8$ across the $ = $ sign we get:
$ \Rightarrow 2y = 12 - 8$
On simplifying we get:
$ \Rightarrow 2y = 4$
On rearranging the terms, we get:
$ \Rightarrow y = \dfrac{4}{2}$
Therefore $y = 2$, is the value of $y$.

Therefore, the values of $x$ and $y$ are $2$ and $2$ respectively.

Note: To check whether the solution is correct we have to test the values of $x$ and $y$ in equation $(2)$
On substituting the values in the left-hand side of the equation, we get:
 $ \Rightarrow 2 + 2(2)$
On simplifying we get:
 $ \Rightarrow 2 + 4$
Which is equal to $6$, which is the right-hand side, therefore the solution is correct.
It is to be remembered that in any given equation multiplying or dividing the equation by a specific constant doesn’t change the value of the equation.
In the given question we had two variables which are $x$ and $y$, therefore they can be solved by using elimination, where there are more than three variables; matrix is used to solve them.