Pair Of Linear Equations In Two Variables Class 10 Chapter Notes

Key Points to Remember

• A linear equation in two variables (x and y) is of the form ax + by + c = 0, where a, b, c are real numbers and a, b are not both zero.
• A pair of linear equations is a system of two such equations: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0.
• The solution to a pair of linear equations is a pair of values (x, y) that satisfies both equations simultaneously.
• Graphically, each linear equation represents a straight line. The solution(s) correspond to the point(s) of intersection of these lines.
• There are three possible graphical outcomes: intersecting lines (unique solution), parallel lines (no solution), or coincident lines (infinitely many solutions).
• A pair of equations is consistent if it has at least one solution (intersecting or coincident lines).
• A pair of equations is inconsistent if it has no solution (parallel lines).
• Algebraic methods include Substitution, Elimination, and Cross-Multiplication.
• Converting word problems into algebraic equations is a crucial skill in this chapter.

Essential Definitions

Linear Equation in Two Variables

An equation of the form Ax + By + C = 0, where A, B, and C are real numbers, and A and B are not both zero. It represents a straight line when graphed.

Pair of Linear Equations

A system consisting of two linear equations involving the same two variables, e.g., a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0.

Solution of a Pair of Linear Equations

A pair of values (x, y) that satisfies both equations in the pair simultaneously. Graphically, it is the point of intersection of the two lines.

Consistent System

A system of linear equations that has at least one solution (either a unique solution or infinitely many solutions).

Inconsistent System

A system of linear equations that has no solution. The lines representing the equations are parallel.

Unique Solution

When a pair of linear equations has exactly one common solution. Graphically, the lines intersect at a single point.

Infinitely Many Solutions

When a pair of linear equations has an unlimited number of common solutions. Graphically, the two lines are coincident (one lies entirely on top of the other).

Graphical Interpretation and Consistency

Understanding the graphical representation of a pair of linear equations is fundamental. Each linear equation, say ax + by + c = 0, represents a straight line on a Cartesian coordinate plane. When we consider a pair of linear equations, we are essentially looking at the relationship between two such lines.

There are precisely three possibilities for two lines in a plane:

1. Intersecting Lines: The two lines cross each other at exactly one point. This point of intersection is the unique solution to the pair of equations. In this case, the system is consistent.
2. Parallel Lines: The two lines never intersect; they remain equidistant from each other. Since there is no common point, there is no solution to the pair of equations. Such a system is called inconsistent.
3. Coincident Lines: The two lines lie exactly on top of each other, meaning every point on one line is also on the other. This implies they share infinitely many solutions. This system is also consistent, and sometimes referred to as 'dependent consistent'.

These graphical conditions can be directly related to the coefficients of the equations. For a pair of linear equations:
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0

The relationships are crucial for quickly determining the nature of solutions without graphing:

• For unique solution (intersecting lines): a₁/a₂ ≠ b₁/b₂
• For no solution (parallel lines): a₁/a₂ = b₁/b₂ ≠ c₁/c₂
• For infinitely many solutions (coincident lines): a₁/a₂ = b₁/b₂ = c₁/c₂

These ratios are vital for quickly classifying the system and predicting the number of solutions, saving time in exams.

Summary of Graphical and Algebraic Conditions

Algebraic Methods for Solving Linear Equations

1. 1. Substitution Method — This method involves expressing one variable in terms of the other from one equation and substituting this expression into the second equation.
2. Steps for Substitution Method: —
3. 2. Elimination Method — This method aims to eliminate one variable by making its coefficients equal (or additive inverses) in both equations and then adding or subtracting the equations.
4. Steps for Elimination Method: —
5. 3. Cross-Multiplication Method — A direct method derived from the elimination method, particularly useful for quickly finding solutions once the formula is memorized. Ensure equations are in the standard form: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0.
6. Formula and Steps for Cross-Multiplication Method: —

Exam Tip: Mastering Word Problems

Many questions in exams involve word problems that need to be translated into a pair of linear equations. This step is often where students make errors.

Strategy:

1. Read Carefully: Understand what quantities are unknown and what relationships are given.
2. Assign Variables: Let the two unknown quantities be x and y (e.g., 'Let the speed of the boat in still water be x km/h and the speed of the stream be y km/h').
3. Formulate Equations: Translate each distinct piece of information or condition into a separate linear equation. Look for keywords like 'sum', 'difference', 'twice', 'half', 'more than', 'less than', 'ratio'.
4. Check Units: Ensure all units are consistent.
5. Solve and Verify: Solve the system using any algebraic method and then check if your solution makes sense in the context of the original word problem.

Quick Revision Check

• Q: What is the graphical representation of a pair of linear equations that has no solution? A: Parallel lines.
• Q: State the condition for a pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to have infinitely many solutions. A: The condition is a₁/a₂ = b₁/b₂ = c₁/c₂.
• Q: If you solve a pair of linear equations and find 0 = 5, what does this imply about the solution? A: It implies that the system has no solution (inconsistent system), and the lines are parallel.
• Q: Which algebraic method would you prefer if one equation already expresses 'y' in terms of 'x' (e.g., y = 2x - 3)? A: The Substitution Method would be most efficient in this case.

Frequently Asked Questions