Question

How do you solve $ 2x + y = 5 $ and $ y = 3x + 2 $ using substitution?

Answer 1

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

Complete step-by-step solution:
Substitution method can be defined as a way to solve a linear system of equations, this method works by substituting one $ y $ -value with the other. The steps to solve by using substitution method are:
First we need to solve one equation for one of its variables.
Now we need to substitute this expression into the other equation and solve it.
Now we need to re-substitute the value into the original equation and we will be able find the corresponding variable.
Given equations are,
 $ 2x + y = 5, $ and $ y = 3x + 2 $ ,
Now first we have to solve one equation to one of its variable,
Here second equation is in variable $ y $ ,
So we can now substitute the second equation in the first equation, we get
 $ 2x + y = 5 $ ,
Now substitute the $ y $ in the equation we get,
 $ \Rightarrow 2x + 3x + 2 = 5 $ ,
Now adding the like terms,
 $ \Rightarrow 5x + 2 = 5 $ ,
Now taking constant term to the right hand side,
 $ \Rightarrow 5x = 5 - 2 $ ,
Now simplifying we get,
 $ \Rightarrow 5x = 3 $ ,
 Now multiplying both sides with 5 we get,
 $ \eqalign{
  & y = 3x + 2 \cr
  & \Rightarrow x = \dfrac{3}{5}y = \dfrac{{19}}{5}\therefore \cr} $ $ \Rightarrow \dfrac{{5x}}{5} = \dfrac{3}{5} $ ,
Now simplifying we get,
 $ \Rightarrow x = \dfrac{3}{5} $ ,
Now substituting the value of $ x $ in the second equation, we get,
 $ y = 3x + 2 $ ,
Now we know that $ x = \dfrac{3}{5} $ , now substituting we get,
 $ \Rightarrow y = 3\left( { \dfrac{3}{5}} \right) + 2 $ ,
Now simplifying we get,
 $ \Rightarrow y = \dfrac{9}{5} + 2 $ ,
Now adding by taking LCM we get,
 $ \Rightarrow y = \dfrac{{9 + 10}}{5} $ ,
Now simplifying we get,
 $ \Rightarrow y = \dfrac{{19}}{5} $ .
The value of $ x $ and $ y $ are, $ x = \dfrac{3}{5} $ and $ y = \dfrac{{19}}{5} $ .

When the given equations $ 2x + y = 5 $ and $ y = 3x + 2 $ are solved using substitution method, we get the value of $ x = \dfrac{3}{5} $ and $ y = \dfrac{{19}}{5} $ .

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.