Question

Solve by substitution: $x+y=2$ and $3x+2y=5$?

Answer 1

In this problem we need to solve the given linear equations by using the substitution method. In the substitution method we will find the value of one variable in terms of another variable from any one of the given linear equations. So here we will calculate the value of $x$ in terms of $y$ from the given equation $x+y=2$ by using some basic mathematical operations. Now we will substitute the calculated $x$ in the other given equation which is $3x+2y=5$ and simplify the equation by using some mathematical operation to get the value of $y$. After getting the value of $y$ we will use any one of the given equations to find the value of $x$.

Complete step-by-step solution:
Given equations are $x+y=2$, $3x+2y=5$.
Considering the equation $x+y=2$. Subtracting $y$ from both sides of the equation, then we will get
$\begin{aligned} & x+y-y=2-y \\\ & \Rightarrow x=2-y \\\ \end{aligned}$
Now substituting the above calculated $x$ value in the another given equation $3x+2y=5$, then we will have
$3\left( 2-y \right)+2y=5$
Using the distribution law of mathematics over subtraction in the above equation, then we will get
$\begin{aligned} & 3\times 2-3\times y+2y=5 \\\ & \Rightarrow 6-3y+2y=5 \\\ \end{aligned}$
Simplifying the above equation by using basic mathematical operations, then we will have
$\begin{aligned} & 6-y=5 \\\ & \Rightarrow y=6-5 \\\ & \therefore y=1 \\\ \end{aligned}$
From the above equation we have the value of $y$ as $1$.
Now substituting the value of $y$ in the given equation $x+y=2$ and simplifying it by using basic mathematical operations, then we will get
$\begin{aligned} & x+1=2 \\\ & \Rightarrow x=2-1 \\\ & \therefore x=1 \\\ \end{aligned}$
Hence the solution for the given equations $x+y=2$ and $3x+2y=5$is $x=1$, $y=1$.

Note: For solving linear equations we have different methods like matrix method, graphical method, substitution method. In this problem they have particularly mentioned to use the substitution method, so we have used substitution method only. However, the linear equations can be easily solved by graphical method. When we plot the graph of the given equation, then the graph will be

From both the methods we got the solution as $x=1$, $y=1$.