Question

How do you solve $2x + y = 0$ and $x - y = 1$ by using a substitution method?

Answer 1

Hint: In this question we have to solve the equations by using substitution method, and in this method first convert the equation in terms of only one variable by using one of its equation and substitute the obtained value in one of the equation to get the other value.

Complete step by step answer:
Substitution method can be defined as a way to solve a linear system of equations, this method works by substituting one $x$-value with the other.
Given equations are,
$2x + y = 0$ and $x - y = 1$ ,
Now let’s take first equation i.e., $2x + y = 0$, convert the equation in terms of $y$, we get,
First subtract 2x from both sides of the equation we get,
$ \Rightarrow 2x + y - 2x = 0 - 2x$,
Now simplifying we get,
$ \Rightarrow y = - 2x$,
So we can now substitute the first equation i.e., $y = - 2x$ in the second equation, Now by substituting the equation we get,
$ \Rightarrow x - \left( { - 2x} \right) = 1$,
Now simplifying we get,
$ \Rightarrow x + 2x = 1$,
Now simplifying we get,
$ \Rightarrow 3x = 1$,
Now divide both sides with 3 we get,
$ \Rightarrow \dfrac{{3x}}{3} = \dfrac{1}{3}$,
Now simplifying we get,
$ \Rightarrow x = \dfrac{1}{3}$,
Now substitute the $x = \dfrac{1}{3}$ in the first equation, Now by substituting the equation we get,
$ \Rightarrow 2\left( {\dfrac{1}{3}} \right) + y = 0$,
Now subtracting $\dfrac{2}{3}$ from both sides of the equation, we get,
$ \Rightarrow \dfrac{2}{3} + y - \dfrac{2}{3} = 0 - \dfrac{2}{3}$,
Now simplifying we get,
$ \Rightarrow y = - \dfrac{2}{3}$,

$\therefore $When the given equations $2x + y = 0$ and $x - y = 1$ solve using substitution method, we get the value of $x$ and $y$ as, $\dfrac{1}{3}$and $\dfrac{{ - 2}}{3}$.

Note: The substitution method is easy to and it works because we have equality in the objects we are substituting for any given equation. If A=B, then we would be able to use B whenever we could use A. So, when we have an equation we are free to do operations to both sides of the equation. This method is better because it makes solving equations much easier, also depending on the equation, this method involves less work and calculation. This method is the most useful system of two equations to solve two unknowns.