Graphical Methods of Solving Pair of Linear Equations in Two Variables

A system of linear equations is just a pair of two lines that may or may not intersect. The graph of a linear equation is a line. There are various methods that can be used to solve two linear equations, for example, Substitution Method, Elimination Method, etc. An easy-to-understand and beginner-friendly way to solve a System of linear equations is by using the graphical methods in which we draw the graph of the equations and try to find that intersection point. First, let's see some linear equation and their graphs. 

Linear Equations and Graphs

For placing the two linear equations on a graph, draw them on the graph and see where and if they intersect. The lines intersecting on the graph give intersecting points known as the solution of the linear equations. Let's look at a pair of linear equations and their graphs from real-life world problems.  

Question 1: Roman went to a stationery shop and purchased 2 pencils and 3 erasers for Rs 9. His friend Sonam saw the new variety of pencils and erasers with Roman, and she also bought 4 pencils and 6 erasers of the same kind for Rs 18. Represent this situation algebraically and graphically.

Solution:

Let's say cost of one pencil is Rs.x and cost of one eraser be Rs. y. Then the equations can be formulated as, 

2x + 3y = 9, 

4x + 6y = 18, 

Let's plot both of these on graph,

By putting them in intercept form, 

\frac{2x}{9} + \frac{3y}{9} = 1

\frac{4x}{18} + \frac{6y}{18} = 1

The graph of both equations looks as follows, 

Both of these lines have same equations, and thus they coincide.

Question 2: A coach of the cricket team bought 3 bats and 6 balls for Rs 3900. Later, he bought another bat and 3 more balls of the same kind for Rs 1300. Represent this situation algebraically and geometrically.

Solution:

Let's say that cost of each bat is Rs. "x" and Balls is Rs. "y". 

3x + 6y = 3900 

x + 3y = 1300

Let's make graph of these equations, it can be made through any of the forms that were taught, but we will stick to intercept form 

\frac{3x}{3900} + \frac{6y}{3900} = 1

\frac{x}{1300} + \frac{3y}{1300} = 1

Now the graphs of both equations come out to be, 

We can see that these lines intersect at one point. 

Graphical Method of Solving Linear Equations

In this method, we make graph of both the equations and then try to find the intersections. Now in graphing and finding the possible intersections, there can be three cases that can occur

1. Unique solution
2. No Solution
3. Infinitely Many Solutions

Unique Solution 

A pair of linear equations in two variables, which has a solution, is called a consistent pair of linear equations. In such cases, both the lines intersect each other. 

For example: 

Question 1: Solve the given pair of equations with the graphical method. 

x + 3y = 6 

2x - 3y = 12

Solution:

We need to first plot both of them on to find the solution. Let's bring them in intercept form, 

\frac{x}{6} + \frac{3y}{6} = 1

\frac{2x}{12} - \frac{3y}{12} = 1

Both of these equations can now be plotted on graph. 

We can see on graph that both these lines intersect on (6,0) so this is the solution to this pair of linear equations. 

Question 2: Ragini went to a shop to buy some pants and skirts. When her friends asked her how many of each she had bought, she answered, “The number of skirts is two less than twice the number of pants purchased. Also, the number of skirts is four less than four times the number of pants purchased”. Help her friends to find how many pants and skirts Ragini bought.

Solution:

Let's say there are "x" number of pants and "y" number of Skirts. Then following equations are formed. 

y = 2x - 2

 y = 4x - 4

Let's draw graphs for both the equations, 

From graph, we can see that these lines intersect at (1,0). So this is the solution for this system of equations 

x = 1 and y = 0 

That means that Ragini brought 1 pant and 0 skirts. 

No Solution 

A pair of linear equations in two variables, which has no solution, is called an inconsistent pair of linear equations. In such cases both the lines are parallel, so they never intersect. 

For example: 

Question: Draw the linear equations on the graph for the given two equations,

7x- 9y = 4

7x- 9y = 12

Solution: 

The graph for the given two equations will look like,

As it is clear that the lines are not intersecting each other at any point, they are parallel in nature. Hence, there is no solution for the given linear equations.

Infinitely Many Solutions 

A pair of linear equations in two variables, which have infinitely many solutions, is called a dependent pair of linear equations. In such cases, both the lines coincide with each other. 

For example: 

Let's generalize the above cases, 

a₁x + b₁y + c₁ = 0 

a₂x + b₂y + c₂ = 0

These equations are, 

1. Intersecting if, \frac{a_1}{a_2} \ne \frac{b_1}{b_2}
2. Coincident if, \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} 
3. Parallel if, \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}

Question: Draw the linear equations on the graph, also find the solution for the two linear equations.

8x+13y = 17

16x+26y = 34

Solution:

The two equations are not only parallel, they are overlapping each other, hence all the points are intersecting each other. Therefore, There exist infinitely many solutions for the given linear equations.

Practice Problems - Graphical Methods of Solving Pair of Linear Equations in Two Variables

Question 1 : Solve: 3u + 2t = 8 ; 5u + 9t = 2 using Graphical method.

Question 2 : Solve 7p + 4q = 7 ; 8p + 5q = 5 using Graphical method.

Question 3 : Solve the following System of Linear Equations : x - 2y = 4 ; 2x - 4y = 8

Question 4 : 2x + 3y = 6
2x + 3y = 12

Some Real Life Applications of Linear Equations

1. Budgeting

Linear equations can model income and expenses. For example :

y = 500x + 200

This could represent monthly expenses where $500 is the rent and $200 is other fixed costs.

2 Distance, Speed, and Time

Linear equations are used in calculating Distance travelled :

d = vt

where d is distance, v is speed(or velocity) , and t is time.

3 Business and Economics

Linear equations model profit and loss :

P = R − C

where, P is profit, R is revenue, and C is cost.

4 Chemistry

In chemistry, linear equations can represent reaction rates and concentrations.

Conclusion

Understanding Linear Equations and their graphs is fundamental in various fields such as Finance, Travel, Business, and Science. The graphical method is a visual way to solve systems of linear equations, which is very useful for finding the point of intersection as the solution.

Also read :

• What is a System of linear equations
• Linear Equations in One Variable
• Difference between Linear and Non-Linear Equations
• Solve Linear Equations with Variable on both sides
• Introduction to Two-Variable Linear Equations in Straight Lines