Chapter 7, Problem 2CT

To determine

To solve: the system by the method of substitution and checking the solution graphically.

Answer to Problem 2CT

The total number of solutions are three, that is, (1,0), (0,−1) and (2, 1).

Explanation of Solution

Given information :

  $\begin{array}{l}y=x-1\\ y={\left(x-1\right)}^3\end{array}$

System of equations are given as,

  $\begin{array}{l}y=x-1\\ y={\left(x-1\right)}^3\end{array}$

Taking 1^st equation and substituting it in the 2^nd equation,

So,

  $\begin{array}{c}x-1={\left(x-1\right)}^3\\ {\left(x-1\right)}^2=0\\ x=1\end{array}$

Further, putting ‘ x’ in first equation,

  $\begin{array}{c}y=x-1\\ y=1-1\\ =0\end{array}$

So,

The solution of system of equations is $\left(1,0\right)$ .

Now, from 1^st equation, finding x and substituting it in 2^nd equation,

  $\begin{array}{c}\Rightarrow y=x-1\\ \Rightarrow x=y+1\end{array}$

Now, substituting,

  $\begin{array}{c}y={\left(y+1-1\right)}^3\\ y=y^3\\ y^2=1\\ y=\pm 1\end{array}$

The other solutions becomes (0,−1) and (2,1).

Therefore, the total number of solutions are three, that is, (1,0),(0,−1) and (2, 1).

The above solution can be checked after taking reference from the graph given below,