medium difficulty6 min

Graphical representation of a pair of linear equations

Graphical representation means drawing both linear equations on the same Cartesian plane and using their position to understand the solution.

In one line: Two equations, one graph, one answer pattern: meet once, never meet, or fully overlap.

Practice This Concept

Understand it

Each linear equation in two variables forms a straight line. When both lines are drawn together, their meeting point, overlap, or separation tells us how many solutions the pair has. This visual method is useful for understanding solution type quickly and for checking algebraic answers.

Example

If one line crosses another at one point, that point gives the unique solution. If the lines are the same line, every point on it is a solution. If they are parallel, there is no common point.

Formula and rule

For a line, at least two points are enough to draw its graph accurately, and the solution of a pair is read from the relation between the two lines.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1For x + y = 4 and x - y = 2, plot points (4,0), (0,4) for the first line and (2,0), (0,-2) for the second line.
2The lines intersect at (3,1), which is the solution.

Method you can reuse

1Rewrite each equation in a form easy to plot.
2Find two points for each equation.
3Plot both sets of points on the same axes.
4Draw straight lines.
5Read the common point or observe if lines are parallel or coincident.

Sketch it in your answer

Use one coordinate plane, clear scale, labelled axes, and straight lines drawn through accurately plotted points. The common point, if any, should be clearly marked.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What does the common point of two lines on a graph tell you?

Think first, then show answer

The common point gives the solution of the pair. Its coordinates satisfy both equations, so that point is the ordered pair solution of the two lines.

Common wrong answer

A common wrong answer is to draw the lines on separate graphs. That cannot show the relation between the two equations.

Don't mix this up

Students often draw rough lines without using enough scale or without marking intercepts correctly, which gives the wrong solution type.

Exam tip

Mark at least two correct points for each line, then draw a neat straight line through them on the same graph.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

The exam may ask you to draw the graph, identify the type of solution, or read the solution from the intersection point.

1
Enough for 1 mark
Graphical representation means drawing both linear equations on the same Cartesian plane and using their position to understand the solution.
2
Add for 2 marks
Example: If one line crosses another at one point, that point gives the unique solution. If the lines are the same line, every point on it is a solution. If they are parallel, there is no common point.
3
Add for 3 marks
Each linear equation in two variables forms a straight line.

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