Question

How do you solve the system of \[0.4x-0.1y=2,0.2x+0.5y=1\]?

Answer 1

Hint: To solve a linear equation in two variables, we need to follow some steps in the following order:
Step 1: Choose any of the two-equation, and find the relationship between the two variables.
Step 2: Substitute this relationship in the other equation, to form a linear equation in one variable.
Step 3: Solve the linear equation to get the value of the variable.
Step 4: Substitute this value in one equation to find the value of another variable.

Complete step by step answer:
We are given the system of equations as, \[0.4x-0.1y=2,0.2x+0.5y=1\]. We need to solve them. As we know to solve pairs of linear equations, we need to follow some steps. We will follow them for the given system, as follows,
First step is to choose any equation and find the relation between two variables. We choose the first equation
\[0.4x-0.1y=2\]
Adding \[0.1y\] to both sides of equation, we get
\[\Rightarrow 0.4x-0.1y+0.1y=2+0.1y\]
 Subtracting 2 from both sides of the equation, we get
\[\Rightarrow 0.4x-2=2+0.1y-2\]
\[\Rightarrow 0.4x-2=0.1y\]
Dividing both sides by \[0.1\] of above equation, we get
\[\Rightarrow \dfrac{0.4x-2}{0.1}=\dfrac{0.1y}{0.1}\]
\[\Rightarrow y=\dfrac{0.4x-2}{0.1}\]
In the second step we need to substitute this relationship in the other equation \[0.2x+0.5y=1\]. By doing this, we get
\[\Rightarrow 0.2x+0.5\left( \dfrac{0.4x-2}{0.1} \right)=1\]
In third step, we need to solve the above equation to find the solution for \[x\]
\[\begin{align}
  & \Rightarrow 0.2x+5\left( 0.4x-2 \right)=1 \\
 & \Rightarrow 0.2x+5\times 0.4x-5\times 2=1 \\
 & \Rightarrow 0.2x+2x-10=1 \\
\end{align}\]
Adding 10 to both sides of above equation, we get
\[\begin{align}
  & \Rightarrow 2.2x-10+10=1+10 \\
 & \Rightarrow 2.2x=11 \\
\end{align}\]
dividing both sides by \[2.2\] of above equation, we get
\[\begin{align}
  & \Rightarrow \dfrac{2.2x}{2.2}=\dfrac{11}{2.2} \\
 & \therefore x=5 \\
\end{align}\]
In the fourth step, we substitute this value in any of the equations, to find the solution for another variable, the first equation is \[0.4x-0.1y=2\]. Substituting above value in this equation, we get
\[\begin{align}
  & \Rightarrow 0.4(5)-0.1y=2 \\
 & \Rightarrow 2-0.1y=2 \\
\end{align}\]
Subtracting 2 from both sides, we get
\[\begin{align}
  & \Rightarrow 2-0.1y-2=2-2 \\
 & \Rightarrow -0.1y=0 \\
 & \therefore y=0 \\
\end{align}\]

Hence the solution for the system is \[x=5\And y=0\].

Note: We can check if the solution is correct or not, by substituting these values in the given equation,
For equation \[0.4x-0.1y=2\], substitute \[x=5\And y=0\], we get \[LHS=0.4(5)-0.1(0)=2+0=2=RHS\]
For equation \[0.2x+0.5y=1\], substitute \[x=5\And y=0\], we get\[LHS=0.2(5)+0.5(0)=1+0=1=RHS\]
Hence the solution is correct.