Graphical Method of Solution of a Pair of Linear Equations in Two Variables

The graphical method of solving a pair of linear equations in two variables is a visual approach that helps in understanding the relationship between two equations. By representing each equation as a straight line on a coordinate plane, the solution is obtained at the point where the two lines intersect. This method gives a clear interpretation of solutions and helps in identifying whether a system has a unique solution, no solution, or infinitely many solutions. In this guide you will learn how to solve a pair of linear equations using graphical method with clear steps and examples .

Table of Contents:

• What is a Pair of Linear Equations in Two Variables
• How to Solve a Pair of Linear Equations Graphically
• Types of Solutions
• Solved Examples on Graphical Method of Solving a Pair of Linear Equations in Two Variables
• Common Mistakes to Avoid

What is a Pair of Linear Equations in Two Variables

A linear equation in two variables is an equation of degree one involving two unknowns, typically x and y. Its general form is:  
  a1x+b1y+c1=0    and      a2x+b2y+c2=0
Where a1,b1,c1,a2,b2,c2are real numbers and  a1,b1,a2andb2≠0.

Each linear equation, when plotted on a coordinate plane, gives a straight line. So when you have two equations, you get two lines. The solution, if it exists, is the point (or points) where those two lines meet.

Read more: Important Questions on Linear Equations in Two Variables - Class 10

How to Solve a Pair of Linear Equations Graphically

Here is a standard approach to find solutions of linear equations in two variables graphically:

1. Write both equations in the form y = mx + c or ax + by = c.

This makes it straightforward to find coordinate points by substituting values of x.

2. Create a table of at least two solution points for each equation.

Substitute x = 0 to get the y-intercept, then substitute y = 0 to get the x-intercept. A third point is always good for accuracy.

3. Plot the points on a coordinate plane and draw a straight line through them.

4. Repeat the process for the second equation and draw its line.

5. Observe the lines and identify their relationship.

Observe the lines to check if they are intersecting, overlapping completely, or are parallel to each other. Each outcome has a specific mathematical meaning.

Types of Solutions

When you draw two lines on a graph, only three cases can happen based on which we determine the nature of the solutions. Here are the three types of solutions based on the position of two lines:

• Consistent: When the two lines intersect each other at exactly one point, that point is the solution. It gives one specific value of x and one specific value of y that satisfies both equations. This is called a consistent pair of equations because a unique solution exists.
  The condition for this is: a1a2≠b1b2

• Dependent: The pair of equations is said to be dependent if the two lines coincide. When both equations represent the same line, every single point on that line is a valid solution. There are infinitely many solutions. This happens when one equation is simply a multiple of the other.
  The condition for this is: a1a2=b1b2=c1c2

• Inconsistent: When the two lines never meet, meaning they are parallel, no point satisfies both equations simultaneously. This is called an inconsistent pair of equations. In this case there is no solution.
  The condition for this is :a1a2=b1b2≠c1c2

 

Solved Examples on Graphical Method of Solving a Pair of Linear Equations in Two Variables

Example 1: Unique Solution
Asha and her brother Rohan together have 50 books. Asha has 10 more books than Rohan. Find how many books each of them has using the graphical method.
Solution: Let Asha's books = x and Rohan's books = y.
Asha and her brother Rohan together have 50 books, i.e., x + y = 50
Asha has 10 more books than Rohan, i.e., x - y = 10
Solving the equation x + y = 50 

Solving the equation x - y = 10

When both lines are plotted on a graph.
Both lines intersect at the point (30, 20). This means Asha has 30 books and Rohan has 20 books. 
Check: a₁/a₂ = 1, b₁/b₂ = -1. Since a₁/a₂  ≠ b₁/b₂, the lines intersect and hence have a unique solution.
 
Example 2: Infinitely Many Solutions
Solve the pair of equations graphically: 2x + 3y = 12 and  4x + 6y = 24
Solution: Given 2x + 3y = 12     ----------- (1)
4x + 6y = 24     ----------- (2)
Equation (2) is exactly twice equation (1). Hence, the lines represented by equations (1) and (2) are coincident. Therefore, equations (1) and (2) have infinitely many solutions.
Check: a₁/a₂ = 2/4 = 1/2, b₁/b₂ = 3/6 = 1/2, c₁/c₂ = 12/24 = 1/2. Since all three ratios are equal, the lines coincide.

Example 3: No Solution
Two bus routes in a city are represented by the equations x + 2y = 6 and x + 2y = 10
Solution: Given the two bus routes in a city are represented by the equations x + 2y = 6 and x + 2y = 10.
Solving the equation x + 2y = 6 

Solving the equation x + 2y = 10 

When both lines are plotted on a graph.
Both lines have the same slope (−1/2) but different y-intercepts (3 and 5). They will never meet, no matter how far you extend them on the graph. These are parallel lines.

Check: a₁/a₂ = 1/1 = 1, b₁/b₂ = 2/2 = 1, c₁/c₂ = 6/10 = 3/5. Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel.

Common Mistakes to Avoid

• Plotting only two points and not checking with a third: A single plotting error shifts the entire line. Always find three points and use the third as a check.

• Choosing inconvenient values of x: Avoid values that give messy fractions for y. Start with x = 0 and y = 0 to get the intercepts that are easy to plot.

• Drawing the line without extending it: Your lines should go beyond the plotted points and reach the edges of the graph. The intersection might lie outside the range of your points.

• Not labelling the intersection point: Even if you draw a perfect graph, forgetting to write the coordinates of the intersection means the examiner can't award full marks.

• Confusing coincident and intersecting lines: When the ratios a₁/a₂ = b₁/b₂ = c₁/c₂, both equations produce the same line, not two different ones.