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.