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?

$\begin{array}{rcl}3(x+2y)& =& 4(y-x)\\ 2y& =& 7x\end{array}$

Answer 1

Step 1: Given data:

Given equations are

$$\begin{gathered} 3(x+2y)=4(y-x)-----\left(1\right) \\ 2y=7x--------\left(2\right) \end{gathered}$$

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

Equation(1) can be written as

$\begin{array}{rcl}3x+6y& =& 4y-4x\\ & \Rightarrow & 4x+3x=4y-6y\\ & \Rightarrow & 7x=-2y\\ & \Rightarrow & y=\frac{-7x}{2}\end{array}$

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

$$\begin{gathered} x=0\Rightarrow y=0 \\ x=2\Rightarrow y=\frac{-7}{2}\times 2=-7 \\ x=-2\Rightarrow y=\frac{-7}{2}\times -2=7 \\ therefore(0,0),(2,-7)and(-2,7)arepoints \end{gathered}$$

$x$	$0$	$2$	$-2$
$y$	$0$	$-7$	$7$

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

$$\begin{gathered} 2y=7x \\ \Rightarrow y=\frac{7}{2}x \end{gathered}$$

Substituting different values of x to get the values of y as follows

$$\begin{gathered} x=0\Rightarrow y=0 \\ x=2\Rightarrow y=\frac{7}{2}\times 2=7 \\ x=-2\Rightarrow y=\frac{7}{2}\times -2=-7 \\ therefore(0,0),(2,7)and(-2,-7)arepoints \end{gathered}$$

$x$	$0$	$2$	$-2$
$y$	$0$	$7$	$-7$

Step 4: Geometrical representation of points.

From the graph , the two lines intersect at the point$(0,0)$

i.e. the pair of equations have a unique solutions

Step 5: Checking for point $(0,0)$:

$$\begin{gathered} 3(x+2y)=4(y-x) \\ putting,x=0andy=0. \\ \Rightarrow 3(0+2\times 0)=4(0-0) \\ \Rightarrow 0=0 \end{gathered}$$

$$\begin{gathered} 2y=7x \\ putting,x=0andy=0 \\ \Rightarrow 2\times 0=7\times 0 \\ \Rightarrow 0=0 \end{gathered}$$

Hence, the system of equations is consistent.