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?

$$\begin{gathered} \frac{x}{2}=\frac{y}{3} \\ y-x=1 \end{gathered}$$

Answer 1

Step 1: Given data:

Given equations are

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

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

Multiplying(1) throughout by 6,we get

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

substituting different values for x to get the values of y as follows

$$\begin{gathered} x=0\Rightarrow y=0 \\ x=2\Rightarrow y=3 \\ x=-2\Rightarrow y=-3 \\ therefore,(0,0),(2,3)and(-2,-30arethepoints. \end{gathered}$$.

$x$	$0$	$2$	$-2$
$y$	$0$	$3$	$-3$

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

Writing the eqn in terma of x,

$y=x+1$

substituting different values for x to get the values of y as follows

$$\begin{gathered} x=0\Rightarrow y=1 \\ x=1\Rightarrow y=2 \\ x=-1\Rightarrow y=0 \\ therefore,(0,1),(1,2)and(-1,0)arethepoints. \end{gathered}$$

$x$	$0$	$1$	$-1$
$y$	$1$	$2$	$0$

Step 4: Geometrical representation of points.

From the graph, it is clear that the system of equations have unique solution at point $(2,3)$.

Step 5: Checking for the point $(2,3)$:

$$\begin{gathered} \frac{x}{2}=\frac{y}{3} \\ putting,x=2andy=3 \\ \Rightarrow \frac{2}{2}=\frac{3}{3} \\ \Rightarrow 1=1 \\ \Rightarrow 1-1=0 \\ \Rightarrow 0=0 \end{gathered}$$

$$\begin{gathered} y-x=1 \\ putting,x=2andy=3 \\ \Rightarrow 3-2=1 \\ \Rightarrow 1=1 \\ \Rightarrow 1-1=0 \\ \Rightarrow 0=0 \end{gathered}$$

Hence, the system has a unique solution and is consistent.