📘 Pair of Linear Equations in Two Variables – Class 10 Maths
Pair of Linear Equations in Two Variables is an important chapter in CBSE Class 10 Mathematics. It explains how two linear equations containing two variables can be solved simultaneously to find the values of unknown quantities.
In this chapter, students learn graphical representation, different algebraic methods, consistency of equations and real-life applications. Understanding these concepts will also help you solve CBSE board examination questions confidently.
1. 📖 What is a Pair of Linear Equations?
A linear equation in two variables is an equation that can be written in the form:
Here, x and y are variables, while a, b and c are real numbers, where a and b are not both zero.
A pair of linear equations consists of two such equations involving the same two variables.
a₂x + b₂y + c₂ = 0
The solution of a pair of linear equations is an ordered pair (x, y) that satisfies both equations simultaneously.
2. 📊 Graphical Representation of Linear Equations
A pair of linear equations can be represented graphically by drawing two straight lines on the same coordinate plane. The position of these lines determines the number of solutions.
Three Possible Cases
| Position of Lines | Number of Solutions | Nature |
|---|---|---|
| Intersecting lines | One unique solution | Consistent |
| Parallel lines | No solution | Inconsistent |
| Coincident lines | Infinitely many solutions | Dependent and consistent |
✏️ Case 1: Intersecting Lines
When two lines intersect at exactly one point, the coordinates of their intersection represent the unique solution of the equations.
✏️ Case 2: Parallel Lines
When two lines are parallel, they never intersect. Therefore, there is no common solution satisfying both equations.
✏️ Case 3: Coincident Lines
When two lines lie exactly on top of each other, they have infinitely many common points. Consequently, the equations have infinitely many solutions.
3. 🔍 Consistency of a Pair of Linear Equations
The consistency of a pair of linear equations depends on whether the equations have a common solution or not.
- Consistent Equations: A pair of equations having at least one solution.
- Inconsistent Equations: A pair of equations having no solution.
- Dependent Equations: A pair of equations having infinitely many solutions because both equations represent the same line.
Conditions for Number of Solutions
Consider the general pair of linear equations:
a₂x + b₂y + c₂ = 0
The following conditions determine the nature of their solutions:
| Condition | Number of Solutions | Nature |
|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | One | Consistent |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Infinitely many | Dependent and consistent |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | None | Inconsistent |
4. 🧮 Substitution Method
The substitution method is an algebraic technique used to solve a pair of linear equations. In this method, we express one variable in terms of the other variable and substitute its value into the second equation.
Steps Involved:
- Take one equation and express one variable in terms of the other.
- Substitute this expression into the second equation.
- Solve the resulting equation to find the value of one variable.
- Substitute this value into either original equation to find the other variable.
Example:
Solve the following equations:
x - y = 1
Step 1: From the first equation:
Step 2: Substitute x = 5 - y into the second equation:
5 - 2y = 1
2y = 4
y = 2
Step 3: Put y = 2 in the first equation:
x = 3
5. ➗ Elimination Method
The elimination method is another algebraic method of solving a pair of linear equations. In this method, we eliminate one variable by adding or subtracting the equations after making the coefficients of that variable equal.
Steps Involved:
- Write both equations in standard form.
- Make the coefficients of one variable equal by multiplying the equations with suitable numbers.
- Add or subtract the equations to eliminate one variable.
- Find the value of the remaining variable.
- Substitute the value into an original equation to calculate the other variable.
Example:
Solve the following equations:
3x + 2y = 12
Step 1: Multiply the first equation by 3 and the second equation by 2.
6x + 4y = 24
Step 2: Subtract the second equation from the first equation.
y = 3
Step 3: Substitute y = 3 into the first equation.
2x + 9 = 13
2x = 4
x = 2
6. ✖️ Cross Multiplication Method
The cross multiplication method is an algebraic technique for solving a pair of linear equations. It uses the coefficients of the variables and constant terms to calculate the values of x and y directly.
For the equations:
a₂x + b₂y + c₂ = 0
The formula is:
Therefore:
y = (c₁a₂ - c₂a₁)/(a₁b₂ - a₂b₁)
7. 🌍 Applications of Pair of Linear Equations in Daily Life
Pair of linear equations is not limited to mathematical calculations. It can also be used to solve many real-life problems involving two unknown quantities.
Some Important Applications:
- 💰 Finding the prices of two different products.
- 👨👩👧 Solving age-related problems.
- 🚗 Calculating the speed and distance of vehicles.
- 🪙 Solving problems involving currency and denominations.
- 📚 Finding the number of boys and girls in a group.
- 🛍️ Calculating quantities and costs of different items.
Example of a Word Problem:
The cost of 2 pens and 3 pencils is ₹13, whereas the cost of 3 pens and 2 pencils is ₹12. Find the cost of one pen and one pencil.
Let:
- x = Cost of one pen
- y = Cost of one pencil
According to the question:
3x + 2y = 12
Solving these equations gives:
💡 Cost of one pencil = ₹3
8. 🔄 Equations Reducible to a Pair of Linear Equations
Some equations may not initially appear to be linear equations. However, by substituting suitable expressions or changing variables, they can be converted into a pair of linear equations.
For example:
5/x + 4/y = 2
Put:
The equations become:
5a + 4b = 2
Now, these are a pair of linear equations in two variables and can be solved using algebraic methods.
9. ⭐ Important Points to Remember
- A linear equation in two variables represents a straight line.
- Two intersecting lines have one unique solution.
- Two parallel lines have no solution.
- Two coincident lines have infinitely many solutions.
- A consistent pair has at least one solution.
- An inconsistent pair has no solution.
- Dependent equations have infinitely many solutions.
- The substitution and elimination methods are important algebraic techniques.
- Always verify your final answer by putting the values into both original equations.
10. 📝 Quick Revision Table
| Concept | Important Information |
|---|---|
| General Form | ax + by + c = 0 |
| Graphical Method | Representation using straight lines |
| Intersecting Lines | One solution |
| Parallel Lines | No solution |
| Coincident Lines | Infinitely many solutions |
| Substitution Method | Substitute one variable's expression into the other equation |
| Elimination Method | Eliminate one variable using addition or subtraction |
| Consistency | Existence of at least one solution |
11. 📚 Prepare for Your CBSE Class 10 Maths Examination
Regular practice is the key to mastering Mathematics. Before moving to this chapter, make sure that you have revised the concepts of Real Numbers and Polynomials.
📖 Chapter 1 – Real Numbers
Revise important concepts and strengthen your basics with our Class 10 Mathematics worksheet.
👉 Download Real Numbers Worksheet – Class 10 PDF📘 Chapter 2 – Polynomials
Practise questions related to zeroes of polynomials and the relationship between zeroes and coefficients.
👉 Download Polynomials Worksheet – Class 10 PDF🎯 Study with SkillShala
SkillShala provides educational resources, worksheets, notes and practice material to help CBSE students prepare effectively for their examinations.
👉 Explore More Free Study Resources at SkillShala12. ❓ Frequently Asked Questions (FAQs)
1. What is a pair of linear equations in two variables?
A pair of linear equations consists of two equations involving the same two variables, generally represented as a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0.
2. How many solutions can a pair of linear equations have?
A pair of linear equations can have one unique solution, no solution or infinitely many solutions, depending on the position of their graphs.
3. What is a consistent pair of linear equations?
A pair of linear equations is called consistent if it has at least one common solution. It may have either one solution or infinitely many solutions.
4. What is an inconsistent pair of linear equations?
A pair of linear equations having no common solution is called an inconsistent pair. Graphically, the two lines are parallel.
5. What are dependent linear equations?
Dependent linear equations are equations that represent the same straight line. They have infinitely many common solutions.
6. What is the difference between substitution and elimination methods?
In the substitution method, one variable is expressed in terms of another and substituted into the second equation. In the elimination method, one variable is removed by adding or subtracting the equations.
7. What is the condition for a unique solution?
For two equations in standard form, a₁/a₂ ≠ b₁/b₂ indicates that the lines intersect at one point and the pair has a unique solution.
8. What is the condition for infinitely many solutions?
If a₁/a₂ = b₁/b₂ = c₁/c₂, the two lines are coincident and the equations have infinitely many solutions.
9. What is the condition for no solution?
If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel and the pair has no solution.
10. Why are pair of linear equations important in Class 10 Maths?
This chapter develops algebraic problem-solving skills and helps students understand graphical representations, simultaneous equations and real-life mathematical applications. It is also an important topic for CBSE board examination preparation.
🎓 Final Words
The chapter Pair of Linear Equations in Two Variables becomes easy when you understand the relationship between equations, graphs and their solutions. Practise numerical problems, learn the consistency conditions and solve different types of word problems regularly.
Keep learning and keep practising with SkillShala – your learning partner for CBSE Classes 9 to 12!