Question

How do you solve the system \[5x+2y=20\] and \[x+4y=13\] ?

Answer 1

Hint: In this question, we have to solve two linear equations, to get the value of x and y. So, we will use the Substitution method to solve the problem. We first rewrite the first equation in terms of x and then put it into the second equation, and solve the variable x. therefore, on further calculations, we see that both the variables cancel out and we are left with the constants only, which implies that the system of equations is inconsistent, which is our required answer to the question.

Complete step by step answer:
According to the question, we have to find the values of x and y.
Therefore, we will use the substitution method to do the same.
The equations given in the problem are: \[5x+2y=20\] ---------- (1) and \[x+4y=13\] ------- (2)
So, we will first rewrite the equation (2) in terms of x, which is
\[x+4y=13\]
Now, subtract 4y on both sides of the above equation, we get
\[\Rightarrow x+4y-4y=13-4y\]
As we know, the same terms with opposite signs cancel out each other, therefore we get
\[\Rightarrow x=13-4y\] -------- (3)
Now, we will put the value of equation (3) into equation (1), we get
\[\Rightarrow 5(13-4y)+2y=20\]
Now, we will open the brackets of the above equation, we get
\[\Rightarrow 65-20y+2y=20\]
As we know, the same terms with opposite signs cancel out each other, therefore we get
\[\begin{align}
  & \Rightarrow 65-20y+2y=20 \\
 & \Rightarrow -18y=-45 \\
 & \Rightarrow y=\dfrac{45}{18} \\
 & \Rightarrow y=\dfrac{5}{2} \\
\end{align}\]
Thus, we see that in the above equation, we got the value of x. Therefore, we will find the value of y. From equation (3), we get
\[\begin{align}
  & x=13-4y \\
 & \Rightarrow x=13-4\left( \dfrac{5}{2} \right) \\
 & \Rightarrow x=13-10 \\
 & \Rightarrow x=3 \\
\end{align}\]

Therefore, the solution of the above equation is \[x=3,y=\dfrac{5}{2}\]

Note: Apart from substitution method, there are also other methods to solve linear equations in two variables which are elimination method and graphical method. Elimination method deals with elimination of either of the variables to find the other and graphical method uses representation of equations on graph and then finding their solutions.