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Class 10·Math
Class 10 · Math

Pair of Linear Equations in Two Variables

Many real-world problems involve finding values that satisfy multiple conditions simultaneously.

Feynman Lens

Start with the simplest version: this lesson is about Pair of Linear Equations in Two Variables. If you can explain the core idea to a friend using everyday language, examples, and one clear reason why it matters, you have moved from memorising to understanding.

Many real-world problems involve finding values that satisfy multiple conditions simultaneously. This chapter teaches you how to solve systems of two linear equations in two variables—like finding where two lines intersect. Whether you're balancing a budget with multiple constraints, finding the break-even point in a business, or determining speeds in a chase problem, you're solving pairs of linear equations. Understanding these systems is fundamental to optimization, economics, engineering, and data analysis.

Linear Equations in Two Variables: Lines on a Plane

A linear equation in two variables has the form ax + by + c = 0 (or ax + by = c).

Key insight: Each such equation represents a line on a coordinate plane. Just as chapter-02-polynomials dealt with expressions, this chapter deals with systems of expressions—finding the point where multiple lines meet.

Real-world example:

Three Methods to Solve Pairs of Equations

1. Substitution Method: Replace and Simplify

Express one variable in terms of the other, then substitute into the second equation.

Example:

2. Elimination Method: Cancel Variables

Multiply equations to make one variable's coefficient identical, then subtract to eliminate it.

Example:

3. Cross-Multiplication Method: Fast Track

A formula-based approach using the coefficients directly (useful for specific cases).

Three Possible Outcomes: Lines Can Intersect, Parallel, or Coincide

When you solve a pair of linear equations, you get exactly one of these scenarios:

Intersecting lines (unique solution):

Parallel lines (no solution):

Coincident lines (infinitely many solutions):

Graphical Representation: Seeing the Solution

Drawing both lines on a graph makes the solution obvious:

This visual approach complements algebraic methods and builds geometric intuition.

These equation-solving skills extend to:

Key Formulas and Theorems

Socratic Questions

  1. When you solve a pair of linear equations and get "2 = 3," what does this tell you about the lines? Why is this outcome important?
  1. Can two different lines ever intersect at more than one point? Why or why not?
  1. You're given that a pair of equations is inconsistent. Without solving, what can you say about the lines they represent?
  1. If one equation is x + y = 5 and another is 2x + 2y = 10, how many solutions does this system have? What's special about these two equations?
  1. In a business problem, why might "infinitely many solutions" be as useful as "one unique solution" for decision-making?
🃏 Flashcards — Quick Recall
Term / Concept
What is Pair of Linear Equations in Two Variables?
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Pair of Linear Equations in Two Variables is the central idea of this lesson. Use the chapter examples to explain what it means and why it matters.
Term / Concept
What is Key insight?
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Each such equation represents a line on a coordinate plane. Just as chapter-02-polynomials dealt with expressions, this chapter deals with systems of expressions—finding the point where multiple lines meet.
Term / Concept
What is Real-world example?
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- Akhila rides on the Giant Wheel (3 rupees per ride) and plays Hoopla (4 rupees per game)
Term / Concept
What is Example?
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- From h = w/2, substitute into 3w + 4h = 20
Term / Concept
What is Coincident lines?
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(infinitely many solutions):
Term / Concept
What is Substitution method?
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Solve for one variable, substitute into the other equation
Term / Concept
What is Elimination method?
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Make coefficients equal, add/subtract to eliminate a variable
Term / Concept
What is Condition for unique solution?
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a₁/a₂ ≠ b₁/b₂ (coefficients not proportional)
Term / Concept
What is Condition for no solution?
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a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (parallel lines)
Term / Concept
What is Condition for infinite solutions?
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a₁/a₂ = b₁/b₂ = c₁/c₂ (coincident lines)
Term / Concept
What is the core idea of Linear Equations in Two Variables: Lines on a Plane?
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A linear equation in two variables has the form ax + by + c = 0 (or ax + by = c). Key insight: Each such equation represents a line on a coordinate plane.
Term / Concept
What is the core idea of 1. Substitution Method: Replace and Simplify?
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Express one variable in terms of the other, then substitute into the second equation. Example: - From h = w/2, substitute into 3w + 4h = 20 - Get 3w + 4(w/2) = 20 → 5w = 20 → w = 4, h = 2
Term / Concept
What is the core idea of 2. Elimination Method: Cancel Variables?
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Multiply equations to make one variable's coefficient identical, then subtract to eliminate it.
Term / Concept
What is the core idea of 3. Cross-Multiplication Method: Fast Track?
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A formula-based approach using the coefficients directly (useful for specific cases).
Term / Concept
What is the core idea of Three Possible Outcomes: Lines Can Intersect, Parallel, or Coincide?
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When you solve a pair of linear equations, you get exactly one of these scenarios: Intersecting lines (unique solution): - The lines cross at one point - The pair is called consistent and independent - Example: y = x…
Term / Concept
What is the core idea of Graphical Representation: Seeing the Solution?
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Drawing both lines on a graph makes the solution obvious: - For intersecting lines, the intersection point is the solution - For parallel lines, no point appears on both lines - For coincident lines, every point on the…
Term / Concept
What is the core idea of Connecting to Related Topics?
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These equation-solving skills extend to: - chapter-04-quadratic-equations: Quadratic equations solve similar problems with curved rather than linear relationships - chapter-05-arithmetic-progressions: Arithmetic…
Term / Concept
What is the core idea of Key Formulas and Theorems?
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- Standard form: ax + by + c = 0 - Substitution method: Solve for one variable, substitute into the other equation - Elimination method: Make coefficients equal, add/subtract to eliminate a variable - Condition for…
Term / Concept
What is Akhila rides on the Giant Wheel (3?
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Akhila rides on the Giant Wheel (3 rupees per ride) and plays Hoopla (4 rupees per game)
Term / Concept
What is She plays Hoopla exactly half as many?
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She plays Hoopla exactly half as many times as she rides the Wheel
Term / Concept
What is She spends exactly 20 rupees?
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She spends exactly 20 rupees
Term / Concept
What is The equations are?
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3w + 4h = 20 and h = w/2
Term / Concept
What is From h = w/2, substitute into 3w?
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From h = w/2, substitute into 3w + 4h = 20
Term / Concept
What is Get 3w + 4(w/2) = 20 →?
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Get 3w + 4(w/2) = 20 → 5w = 20 → w = 4, h = 2
Term / Concept
What is Equation 1?
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3w + 4h = 20
Term / Concept
What is Equation 2?
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h = w/2 → −w + 2h = 0
Term / Concept
What is Multiply Equation 2 by 3?
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−3w + 6h = 0
Term / Concept
What is Add to Equation 1?
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10h = 20 → h = 2, then w = 4
Term / Concept
What is The lines cross at one point?
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The lines cross at one point
Term / Concept
What is The pair is called consistent and independent?
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The pair is called consistent and independent
Term / Concept
What is The lines never meet?
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The lines never meet
Term / Concept
What is The pair is called inconsistent?
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The pair is called inconsistent
Term / Concept
What is The lines are identical?
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The lines are identical
Term / Concept
What is The pair is called consistent and dependent?
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The pair is called consistent and dependent
Term / Concept
What is For intersecting lines, the intersection point is?
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For intersecting lines, the intersection point is the solution
Term / Concept
What is For parallel lines, no point appears on?
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For parallel lines, no point appears on both lines
Term / Concept
What is For coincident lines, every point on the?
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For coincident lines, every point on the line is a solution
Term / Concept
What is chapter-04-quadratic-equations?
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Quadratic equations solve similar problems with curved rather than linear relationships
Term / Concept
What is chapter-05-arithmetic-progressions?
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Arithmetic sequences involve solving linear relationships over sequences
Term / Concept
What is chapter-07-coordinate-geometry?
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Points and lines are fundamental to analyzing geometric shapes
40 cards — click any card to flip
📝 Quick Quiz — Test Yourself
When you solve a pair of linear equations and get "2 = 3," what does this tell you about the lines? Why is this outcome important?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
Can two different lines ever intersect at more than one point? Why or why not?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
You're given that a pair of equations is inconsistent. Without solving, what can you say about the lines they represent?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
If one equation is x + y = 5 and another is 2x + 2y = 10, how many solutions does this system have? What's special about these two equations?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
In a business problem, why might "infinitely many solutions" be as useful as "one unique solution" for decision-making?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
Which approach best shows that you understand Pair of Linear Equations in Two Variables?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Key insight?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Real-world example?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Example?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Coincident lines?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Substitution method?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Elimination method?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Condition for unique solution?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Condition for no solution?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Condition for infinite solutions?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Linear Equations in Two Variables: Lines on a Plane?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand 1. Substitution Method: Replace and Simplify?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand 2. Elimination Method: Cancel Variables?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand 3. Cross-Multiplication Method: Fast Track?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Three Possible Outcomes: Lines Can Intersect, Parallel, or Coincide?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Graphical Representation: Seeing the Solution?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Connecting to Related Topics?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Key Formulas and Theorems?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Akhila rides on the Giant Wheel (3?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand She plays Hoopla exactly half as many?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand She spends exactly 20 rupees?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The equations are?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand From h = w/2, substitute into 3w?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Get 3w + 4(w/2) = 20 →?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Equation 1?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Equation 2?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Multiply Equation 2 by 3?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Add to Equation 1?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The lines cross at one point?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The pair is called consistent and independent?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The lines never meet?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The pair is called inconsistent?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The lines are identical?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The pair is called consistent and dependent?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand For intersecting lines, the intersection point is?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
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