Question

How do you solve systems of equations by substitution 2x – 3y = -1 and y = x – 1?

Answer 1

Hint: Assume the given equations as equation (1) and (2) respectively. Now, write the variable ‘y’ in terms of the variable ‘x’ considering the second equation. Substitute this value of y, founded in terms of x, in equation (1) and solve the equation for the value of x. Once the value of x is found, substitute it in equation (2) to find the value of y.

Complete step by step answer:
Here, we have been provided with two equations 2x – 3y = -1 and y = x – 1 and we have been asked to solve this system by the method of substitution. To solve the system of equations means we need to find the values of the variable x and y.
Now, let us assume the two given equations as equation (1) and (2), so we have,
\[\Rightarrow \] 2x – 3y = -1 –(1)
\[\Rightarrow \] y = x – 1 –(2)
The method of substitution states that we have to select one of the two equations and write one of the variables in terms of the other. Now, we have to substitute this obtained value in the non – selected equation to convert it into an equation containing only one variable that can be solved easily.
Now, let us select equation (2), i.e., y = x – 1. Here, we can see that the variable ‘y’ is already written in terms of the variable ‘x’. So, substituting this value of y in equation (1), we get,
\[\begin{align}
  & \Rightarrow 2x-3\left( x-1 \right)=-1 \\
 & \Rightarrow 2x-3x+3=-1 \\
 & \Rightarrow -x+3=-1 \\
 & \Rightarrow -x=-1-3 \\
 & \Rightarrow -x=-4 \\
\end{align}\]
Multiplying both sides with (-1), we get,
\[\begin{align}
  & \Rightarrow \left( -1 \right)\times \left( -x \right)=\left( -1 \right)\times \left( -4 \right) \\
 & \Rightarrow x=4 \\
\end{align}\]
We have obtained the value of x, so substituting it in equation (2), we get,
\[\begin{align}
  & \Rightarrow y=4-1 \\
 & \Rightarrow y=3 \\
\end{align}\]

Hence, the solution of the given system of equations can be given as (x, y) = (4, 3).

Note: One may note that we can also select equation (1) in place of equation (2) and find the value of y in terms of x and proceed. This will also give the same answer. You may check the answer by substituting the values of x and y obtained, in the given equations. If L.H.S and R.H.S turns out to be the same for both the cases then our answer is correct. Remember that we can also solve the question by elimination and cross – multiplication method but we were asked to use the substitution method only.