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# Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (i) $$x + y = 5, 2x + 2y = 10$$(ii)$$x – y = 8, 3x - 3y = 16$$(iii)$$2x + y = 6, 4x - 2y = 4$$(iv)$$2x - 2y – 2 = 0, 4x - 4y – 5 = 0$$

(i) $$x + y = 5, 2x + 2y = 10$$

For equation $$x + y -5 = 0$$, points which lie on the line:

$$\begin{array} {|r|r|}\hline x & 0 & 5 \\ \hline y & 5 & 0 \\ \hline \end{array}$$

For equation $$2x + 2y = 10$$, points which lie on the line:

$$\begin{array} {|r|r|}\hline x & 1 & 2 \\ \hline y & 4 & 3 \\ \hline \end{array}$$

Graph for the above equations is: On plotting the graph we can see that both of the lines coincide.

So, there are infinitely many solutions.

Hence, they are consistent.

(ii) $$x – y = 8, 3x - 3y = 16$$

For equation $$x - y - 8 = 0$$, points which lie on the line:

$$\begin{array} {|r|r|}\hline x & 0 & 8 \\ \hline y & -8 & 0 \\ \hline \end{array}$$

For equation $$3x - 3y = 16$$, points which lie on the line:

$$\begin{array} {|r|r|}\hline x & 0 & 16 \\ \hline y & -16 & 0 \\ \hline \end{array}$$

Graph for the above equations is: On plotting the graph we can see that both of the lines are parallel to each other.

So, there are no solutions.

Hence, they are inconsistent.

(iii) $$2x + y = 6, 4x - 2y = 4$$

For equation $$2x + y - 6 = 0$$, points which lie on the line:

$$\begin{array} {|r|r|}\hline x & 0 & 3 \\ \hline y & 6 & 0 \\ \hline \end{array}$$

For equation $$4x – 2y – 4 = 0$$, points which lie on the line:

$$\begin{array} {|r|r|}\hline x & 0 & 1 \\ \hline y & -2 & 0 \\ \hline \end{array}$$

Graph for the above equations is: On plotting the graph we can see that both of the lines are intersecting each other at exactly one point.

So, there is a unique solution.

Hence, they are consistent.

(iv) $$2x - 2y – 2 = 0, 4x - 4y – 5 = 0$$

For equation $$2x - 2y – 2 = 0$$, points which lie on the line:

$$\begin{array} {|r|r|}\hline x & 2 & 0 \\ \hline y & 0 & -2 \\ \hline \end{array}$$

For equation $$4x – 2y – 4 = 0$$, points which lie on the line:

$$\begin{array} {|r|r|}\hline x & 5 & 0 \\ \hline y & 0 & -5 \\ \hline \end{array}$$

Graph for the above equations is: On plotting the graph we can see that both of the lines are parallel to each other.

So, there is no solution.

Hence, they are inconsistent.