Mathematics Notes for Class 10 Pair Of-Linear-Equations-In-Two-Variables

Pair of Linear Equations in Two Variables

In this chapter, you will learn about pairs of linear equations in two variables, how to represent them, and different methods to solve them. By the end of this chapter, you will be able to solve real-life problems using pairs of linear equations.

Key Concepts

Linear Equation: An equation of the form ax + by + c = 0, where a, b, and c are real numbers, and x and y are variables.
Pair of Linear Equations: Two linear equations in the same two variables.
Solution: The values of x and y that satisfy both equations.

Graphical Representation

Each linear equation in two variables represents a straight line on the coordinate plane. The solution to a pair of linear equations is the point(s) where the two lines intersect.

One Solution: Lines intersect at one point (consistent and independent).
No Solution: Lines are parallel (inconsistent).
Infinitely Many Solutions: Lines coincide (consistent and dependent).

Algebraic Methods of Solving

Substitution Method: Solve one equation for one variable and substitute in the other.
Elimination Method: Add or subtract equations to eliminate one variable.
Cross-Multiplication Method: Use the formula to directly find the values of x and y.

Applications

Solving word problems involving two unknowns.
Finding the intersection point of two lines.
Solving problems related to age, money, distance, and mixtures.

Practice Questions

Solve the pair of equations:
2x + 3y = 12
x - y = 1
If 3 pens and 2 pencils cost ₹19 and 2 pens and 4 pencils cost ₹16, find the cost of one pen and one pencil.
Draw the graphs of x + y = 5 and x - y = 1. Find their point of intersection.
State the condition for a pair of linear equations to have no solution.
Solve using elimination: 4x - 3y = 7, 3x + 4y = 17

Challenge Yourself

Form a pair of linear equations for: "The sum of two numbers is 10 and their difference is 2." Solve them.
A boat covers 30 km downstream in 2 hours and the same distance upstream in 3 hours. Find the speed of the boat in still water and the speed of the stream.

Did You Know?

Linear equations are used in business, science, and engineering to solve real-world problems.
The graphical method helps you visualize the solution of equations.

Glossary

Consistent: Equations that have at least one solution.
Inconsistent: Equations that have no solution.
Dependent: Equations that have infinitely many solutions.
Independent: Equations that have exactly one solution.

Answers to Practice Questions

x = 3, y = 2
Pen = ₹4, Pencil = ₹2.5
Intersection at (3, 2)
If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the pair has no solution.
x = 3, y = 2

Practice solving pairs of linear equations to master this important topic in algebra!

Mathematics Class 10 - Pair Of-Linear-Equations-In-Two-Variables Notes