Question

Solve the following systems of linear equations graphically

(a) If the system has a unique solution, check your answer by substituting in the equation.

(b) If the system is inconsistent or dependent, state which of the two it is?

$$\begin{gathered} y=2-3x \\ x=\frac{2}{3}-\frac{y}{3} \end{gathered}$$

Answer 1

Step 1:Given data:

Given equations are,

$$\begin{gathered} y=2-3x--------\left(1\right) \\ x=\frac{2}{3}-\frac{y}{3}--------\left(2\right) \end{gathered}$$

Step 2: Find the points corresponding to equation (1)

submitting the different values for x to get the values of y

$$\begin{gathered} y=2-3x \\ forx=0\Rightarrow y=2 \\ forx=1\Rightarrow y=2-3\times 1 \\ \Rightarrow y=-1 \\ forx=2\Rightarrow y=2-6 \\ \Rightarrow y=-4 \end{gathered}$$

Therefore$(0,2),(1,-1)(2,-4)$ are the points

Step 3: Find the points corresponding to equation (2)

Dividing eqn (2) throughout by 3,we get

$$\begin{gathered} 3x=2-y \\ ie,y=2-3x \end{gathered}$$

i.e., we get equation(1) and equation (2) are same

Therefore$(0,2),(1,-1)(2,-4)$ are the points for eqn (2) as well.

Step 3: Plotting the points on the graph

The points corresponding to equation(2) is same as that of equation (1).

therefore, the two lines coincide and have infinitely many solutions.

Hence, the system of the equation is dependent.