Premium Online Home Tutors
3 Tutor System
Starting just at 265/hour

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)\(2x - 2y – 2 = 0, 4x - 4y – 5 = 0\)


Answer :

(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:

Graph 1

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:

Graph 2

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:

Graph 3

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:

Graph 4

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.

NCERT solutions of related questions for Pair of linear equations in two variables

NCERT solutions of related chapters class 10 maths

NCERT solutions of related chapters class 10 science