Simply CBSE

Linear Equations in Two Variables — Class 10 Mathematics

Learn to solve linear equations in two variables using multiple methods and graphical representations.

In this chapter, you will learn

  • —Understand the concept of linear equations in two variables and standard form
  • —Represent linear equations graphically on the coordinate plane
  • —Determine if a pair of equations is consistent, inconsistent, or dependent
  • —Solve equations using the graphical method
  • —Apply substitution method for algebraic solutions
  • —Use elimination method for efficient problem solving
  • —Understand cross-multiplication method and its applications
  • —Solve word problems using linear equations

Introduction and Standard Form

Linear Equation in Two Variables: An equation of the form ax + by + c = 0, where a, b, c are real numbers and a, b ≠ 0.

Standard Forms:
• General form: ax + by + c = 0
• Slope-intercept form: y = mx + c (where m is slope)
• Intercept form: x/a + y/b = 1

Examples of Linear Equations in Two Variables:

  • 2x + 3y = 6 (rearrange: 2x + 3y - 6 = 0)
  • x - y + 1 = 0
  • 5x + 2y = 10
  • 3x = y - 4 (rearrange: 3x - y + 4 = 0)

Not linear equations in two variables: x² + y = 5 (has x²), xy + 2 = 0 (has product term)

Solution of Linear Equation: Any pair (x, y) that satisfies the equation.

  • A linear equation in two variables has infinitely many solutions
  • Each solution is represented as an ordered pair (x, y)
  • When plotted, all solutions lie on a straight line

Key Points:

  • Linear equations are of the first degree (power of variables is 1)
  • Two variables mean the equation has two unknowns (x and y)
  • The solution set forms a straight line on the graph

Exam Tip

Recognize linear equations quickly by checking that variables are to the first power only and no product terms exist. Convert to standard form for consistency.

Common Mistake

Students confuse linear with quadratic equations. Remember: linear has degree 1 (x, not x²). Also, a = 0 or b = 0 makes it not a linear equation in two variables.

Graphical Representation

A linear equation in two variables can be represented as a straight line on the coordinate plane (Cartesian plane).

Method to Draw the Graph:

  1. Find at least two solutions by substituting values of x
  2. Plot these points on the coordinate plane
  3. Draw a straight line through these points

Example: Graph 2x + y = 4

  • When x = 0: 2(0) + y = 4 → y = 4. Point: (0, 4)
  • When x = 2: 2(2) + y = 4 → y = 0. Point: (2, 0)
  • When x = 1: 2(1) + y = 4 → y = 2. Point: (1, 2)

Plot these three points and draw a straight line through them.

Finding Intercepts (Easier Method):

  • x-intercept: Put y = 0 and solve for x. This gives point (x, 0)
  • y-intercept: Put x = 0 and solve for y. This gives point (0, y)
  • Example: For 3x + 2y = 6:
    • x-intercept: y = 0 → 3x = 6 → x = 2. Point: (2, 0)
    • y-intercept: x = 0 → 2y = 6 → y = 3. Point: (0, 3)

Key Features of the Graph:

  • The graph is always a straight line
  • Slope = -a/b (for equation ax + by + c = 0)
  • Steeper lines have larger |slope| values
  • Horizontal line: coefficient of x is 0 (e.g., y = 3)
  • Vertical line: coefficient of y is 0 (e.g., x = 5)

Exam Tip

For graphing, always use intercept method as it's faster than finding multiple points. Draw clear graphs with proper scale and label axes. The line represents all solutions.

Common Mistake

Students plot wrong points or don't extend the line. Remember: check at least two points, draw a straight line (not curved), and extend it across the entire visible plane.

Consistency and Consistency of System

When dealing with a pair of linear equations in two variables, we need to determine their relationship:

For two equations: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0

Consistency:

  • Consistent & Independent: Unique solution (lines intersect at one point)
    • Condition: a₁/a₂ ≠ b₁/b₂
    • Graphically: Two distinct intersecting lines
  • Consistent & Dependent: Infinitely many solutions (lines coincide/overlap)
    • Condition: a₁/a₂ = b₁/b₂ = c₁/c₂
    • Graphically: Same line (coincident lines)
  • Inconsistent: No solution (lines are parallel)
    • Condition: a₁/a₂ = b₁/b₂ ≠ c₁/c₂
    • Graphically: Parallel lines (never meet)

Examples:

  • Consistent & Independent: x + y = 4 and x - y = 2
    • Ratios: 1/1 = 1, 1/(-1) = -1. Since 1 ≠ -1, these have unique solution (3, 1)
  • Inconsistent: 2x + 3y = 4 and 4x + 6y = 9
    • Ratios: 2/4 = 1/2, 3/6 = 1/2, but 4/9 ≠ 1/2. So these are parallel (no solution)
  • Consistent & Dependent: x + 2y = 4 and 2x + 4y = 8
    • Second equation is twice the first, so all ratios are 1/2. Infinitely many solutions

Key Points:

  • Check the ratios of coefficients to determine consistency
  • Parallel lines have same slope but different intercepts
  • Coincident lines are exactly the same line

Exam Tip

Always check consistency before solving. If a₁/a₂ = b₁/b₂ = c₁/c₂, there are infinitely many solutions. If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, there's no solution.

Common Mistake

Students don't use the consistency conditions properly. Remember: equal ratios of all three coefficients = dependent system. Only first two equal = parallel (no solution).

Substitution Method

A systematic algebraic method to solve a pair of linear equations by substituting one variable in terms of the other.

Steps in Substitution Method:

  1. From one equation, express one variable in terms of the other
  2. Substitute this expression in the second equation
  3. Solve the resulting equation in one variable
  4. Substitute back to find the other variable
  5. Verify the solution in both original equations

Example: Solve 2x + y = 5 and x - y = 1

  • Step 1: From second equation: x = y + 1
  • Step 2: Substitute in first: 2(y + 1) + y = 5
  • Step 3: 2y + 2 + y = 5 → 3y = 3 → y = 1
  • Step 4: x = 1 + 1 = 2
  • Solution: (2, 1)
  • Verification: 2(2) + 1 = 5 ✓ and 2 - 1 = 1 ✓

When to Use Substitution:

  • When one equation is easy to rearrange (e.g., x + 2y = 5 → x = 5 - 2y)
  • When one variable has coefficient 1 or -1
  • When you need to express one variable explicitly

Key Points:

  • Always verify your solution in both original equations
  • Be careful with signs when rearranging
  • This method works for any consistent system with unique solution

Exam Tip

Always show each step clearly and verify at the end. Choose the equation that makes substitution easiest (look for coefficient 1 or -1).

Common Mistake

Sign errors are common. When you have -y = -2, students write y = -2 instead of y = 2. Also, always verify your answer in BOTH equations.

Elimination Method

A powerful algebraic method that eliminates one variable by adding or subtracting the equations appropriately.

Steps in Elimination Method:

  1. Make the coefficients of one variable equal (multiply equations as needed)
  2. Add or subtract equations to eliminate that variable
  3. Solve the resulting single-variable equation
  4. Substitute back to find the other variable
  5. Verify the solution

Example: Solve 3x + 2y = 11 and 2x - y = 3

  • Step 1: Make y coefficients equal: Multiply second equation by 2
    • Equation 1: 3x + 2y = 11
    • Equation 2: 4x - 2y = 6
  • Step 2: Add equations: 7x = 17 → x = 17/7
  • Step 3: Substitute in second: 2(17/7) - y = 3 → y = -11/7
  • Solution: (17/7, -11/7)

Choosing Which Variable to Eliminate:

  • Choose the variable that requires smallest multipliers
  • Look for coefficients that are already multiples of each other
  • Example: If coefficients are 2 and 3, multiply by 3 and 2 (LCM = 6)

Key Points:

  • Elimination is faster when coefficients are already simple multiples
  • Be careful with signs when multiplying entire equations
  • Add equations to eliminate, subtract equations to eliminate (depending on signs)

Exam Tip

Elimination is often faster than substitution. Choose to eliminate the variable with easier coefficients. Always verify your final answer.

Common Mistake

Sign errors when multiplying equations. If multiplying 2x - y = 3 by 2, it becomes 4x - 2y = 6, not 4x + 2y = 6. Check signs carefully.

Cross-Multiplication Method

A direct formula-based method that gives solutions instantly for certain pair of equations.

Cross-Multiplication Formula:

For equations: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0

x/(b₁c₂ - b₂c₁) = y/(c₁a₂ - c₂a₁) = 1/(a₁b₂ - a₂b₁)

Example: Solve 2x + 3y - 8 = 0 and x - 2y + 3 = 0

Here: a₁ = 2, b₁ = 3, c₁ = -8; a₂ = 1, b₂ = -2, c₂ = 3

  • b₁c₂ - b₂c₁ = 3(3) - (-2)(-8) = 9 - 16 = -7
  • c₁a₂ - c₂a₁ = (-8)(1) - 3(2) = -8 - 6 = -14
  • a₁b₂ - a₂b₁ = 2(-2) - 1(3) = -4 - 3 = -7
  • Therefore: x/-7 = y/(-14) = 1/(-7)
  • x = -7/-7 = 1 and y = -14/-7 = 2
  • Solution: (1, 2)

When to Use Cross-Multiplication:

  • When a quick solution is needed
  • Most reliable for competitive exams
  • Reduces chance of arithmetic errors if formula is applied correctly

Key Points:

  • Equations must be in form ax + by + c = 0
  • If denominator (a₁b₂ - a₂b₁) = 0, system is either inconsistent or has infinite solutions
  • This is a direct method with no intermediate steps

Exam Tip

Learn the cross-multiplication formula pattern well. It's a direct method that avoids algebraic manipulations. Use a systematic approach to avoid calculation errors.

Common Mistake

The biggest mistake is mixing up the numerators. Remember: x uses coefficients of y and c, y uses coefficients of x and c. Practice the pattern carefully.

Word Problems and Applications

Linear equations in two variables are used to solve real-world problems involving two unknowns.

Steps to Solve Word Problems:

  1. Read the problem carefully and identify two unknowns
  2. Assign variables (x, y) to the unknowns
  3. Form equations based on given conditions
  4. Solve using any method (substitution, elimination, cross-multiplication)
  5. Verify that the answer makes sense in the context

Example 1: Age Problem

Problem: The sum of ages of two people is 24 years. Five years ago, the age of one was 3 times the age of the other. Find their current ages.

  • Let their current ages be x and y
  • Equation 1: x + y = 24
  • Equation 2: (x - 5) = 3(y - 5) → x - 5 = 3y - 15 → x = 3y - 10
  • Substitute in equation 1: (3y - 10) + y = 24 → 4y = 34 → y = 8.5
  • x = 24 - 8.5 = 15.5
  • Answer: Ages are 15.5 and 8.5 years

Example 2: Cost Problem

Problem: The cost of 5 pens and 3 pencils is Rs. 71. The cost of 3 pens and 5 pencils is Rs. 65. Find the cost of one pen and one pencil.

  • Let cost of pen = x and cost of pencil = y
  • Equation 1: 5x + 3y = 71
  • Equation 2: 3x + 5y = 65
  • Add equations: 8x + 8y = 136 → x + y = 17
  • Answer: Cost of one pen and one pencil = Rs. 17

Common Problem Types:

  • Age problems: Sum and difference of ages, or relationships at different times
  • Cost/Price problems: Cost of multiple items, total cost
  • Distance/Speed problems: Time, speed, and distance relationships
  • Percentage problems: Two quantities with percentage changes
  • Mixture problems: Two mixtures with different compositions

Key Points:

  • Clearly define variables and their units
  • Translate words into algebraic equations carefully
  • Check if answer is reasonable (ages should be positive, costs should make sense)
  • Always state answer in the context of the problem

Exam Tip

Read the problem twice before starting. Identify all conditions carefully. Common keywords: 'sum' (addition), 'difference' (subtraction), 'times' (multiplication).

Common Mistake

Students set up equations incorrectly. 'The difference is 5' means x - y = 5 or y - x = 5 (not both). 'One is 3 times the other' means y = 3x, not y = x + 3.

Chapter Summary

Linear Equations in Two Variables form the foundation of algebra and are essential for solving real-world problems. This chapter covers:

  • Concepts: Introduction, standard forms, and solution sets
  • Graphical Method: Visual representation as straight lines
  • Consistency: Determining if system has unique, infinite, or no solutions
  • Substitution Method: Express and substitute one variable
  • Elimination Method: Eliminate variables through addition/subtraction
  • Cross-Multiplication: Direct formula-based solution
  • Applications: Solving real-world word problems

Exam Focus: Consistency of systems, all three solving methods, word problems, verification of solutions.

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