Class 10 Mathematics – Pair of Linear Equations | Interactive Visual Revision Notes

📝 Section 1 — Classroom Notes

📐 Topic 1: Linear Equation in Two Variables

🌎 Real World Connection: A shopkeeper sells pens at ₹5 and pencils at ₹2. A customer spends ₹31 buying some of each. Another customer buys twice as many pens and spends ₹38. To find exactly how many of each were bought, we set up two equations with two unknowns and solve them together — that is a pair of linear equations in action.

A linear equation has two variables (x and y).
General form: ax + by + c = 0.
Here a, b, c are real numbers.
a ≠ 0 and b ≠ 0 (both must exist).
Its graph is always a straight line.

📐 Topic 2: Pair of Linear Equations

A pair means two equations together.
Example: 2x + 3y = 12 and x − y = 1.
We need to find one x and one y that works for both.
This (x, y) is the solution.
On a graph, it is where the lines cross.

📐 Topic 3: Consistent and Inconsistent

Consistent = the pair has at least one solution.
If lines intersect at one point = exactly one solution.
If lines overlap completely = infinite solutions.
Inconsistent = the pair has no solution.
This happens when lines are parallel (never meet).

📐 Topic 4: The Ratio Test

For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
If a₁/a₂ ≠ b₁/b₂: lines intersect (one solution).
If a₁/a₂ = b₁/b₂ ≠ c₁/c₂: lines are parallel (no solution).
If a₁/a₂ = b₁/b₂ = c₁/c₂: lines overlap (infinite solutions).
This test tells you the answer without drawing.

📐 Topic 5: Graphical Method

Draw both lines on a graph.
Find at least two points for each line.
Plot the points, draw the lines.
Where they cross = the solution.
Read the (x, y) value at the crossing point.

📐 Topic 6: Substitution Method

Step 1: From one equation, write x in terms of y (or y in terms of x).
Step 2: Put this into the other equation.
Step 3: Solve for the one remaining variable.
Step 4: Put the answer back to find the other.
Works best when one variable has coefficient 1.

📐 Topic 7: Elimination Method

Goal: make the coefficients of one variable equal.
Multiply one or both equations to match.
Add or subtract the equations.
One variable disappears!
Solve for the remaining one, then find the other.

📐 Topic 8: When to Use Which Method

Graphical: good for understanding, not exact.
Substitution: best when a variable has coefficient 1.
Elimination: works well with any coefficients.
All methods give the same answer.
Pick the one that looks easiest!

📐 Topic 9: Word Problems

Read the problem carefully.
Choose two variables (let x = ..., y = ...).
Write two equations from the clues.
Solve using any method.
Check your answer makes sense in the story.

📐 Topic 10: Special Cases

If you get 0 = 0: infinite solutions (lines overlap).
If you get 0 = 5 (impossible): no solution (parallel).
If you get x = number: exactly one solution.
Always check by putting answers back.
A pair with infinite solutions is called dependent.

🎨 Section 2 — Easy Illustrations

Picture 1: Three Types of Line Pairs

Three possible outcomes when two lines meet (or don't).

Picture 2: Graphical Solution

The lines x + y = 7 and x − y = 1 cross at (4, 3). That is the solution!

Picture 3: The Ratio Test Decision Tree

The ratio test tells you the answer without solving the equations.

Picture 4: Substitution Method (Step-by-Step)

Four simple steps: rearrange, substitute, solve, find the other.

Picture 5: Elimination Method (Step-by-Step)

Subtract the equations so one variable disappears. Then solve!

✏️ Section 3 — Worked Examples

Example 1: Solve by Substitution

Question: Solve 3x + y = 10 and x − 2y = −1.

1 From equation 2: x = 2y − 1.

2 Put into equation 1: 3(2y − 1) + y = 10.

3 6y − 3 + y = 10 → 7y = 13.

4 y = 13/7.

5 x = 2(13/7) − 1 = 26/7 − 7/7 = 19/7.

6 Check: 3(19/7) + 13/7 = 57/7 + 13/7 = 70/7 = 10 ✓

x = 19/7, y = 13/7

Example 2: Solve by Elimination

Question: Solve 4x + 3y = 25 and 2x − 3y = −1.

1 Coefficients of y are +3 and −3. Add both equations.

2 (4x + 2x) + (3y − 3y) = 25 + (−1).

3 6x = 24 → x = 4.

4 Put in equation 1: 4(4) + 3y = 25 → 3y = 9 → y = 3.

5 Check: 2(4) − 3(3) = 8 − 9 = −1 ✓

x = 4, y = 3

Example 3: Word Problem

Question: Ravi bought 5 notebooks and 3 pens for ₹115. Meena bought 3 notebooks and 5 pens for ₹105. Find the cost of each.

1 Let notebook = x, pen = y.

2 Equations: 5x + 3y = 115 and 3x + 5y = 105.

3 Multiply eq1 by 5 and eq2 by 3: 25x + 15y = 575 and 9x + 15y = 315.

4 Subtract: 16x = 260 → x = 16.25.

5 5(16.25) + 3y = 115 → 3y = 33.75 → y = 11.25.

6 Check: 3(16.25) + 5(11.25) = 48.75 + 56.25 = 105 ✓

Notebook = ₹16.25, Pen = ₹11.25

Term	What It Means	Key Fact
Linear Equation	ax + by + c = 0	Straight line graph
Pair of Equations	Two equations, two unknowns	Find x and y
Consistent	Has solution(s)	Lines intersect or overlap
Inconsistent	No solution	Parallel lines
Dependent	Infinite solutions	Same line
Graphical Method	Draw and find crossing	Visual but slow
Substitution	Replace one variable	Best if coeff = 1
Elimination	Add/subtract to remove	Works for any pair
Ratio Test	Compare a₁/a₂ and b₁/b₂	Quick check
Unique Solution	a₁/a₂ ≠ b₁/b₂	Lines cross once

📄 Section 5 — Take-Home Handout

📝 A. Fill in the Blanks

1. A linear equation in two variables has the form _________.
2. If two lines intersect, the pair is called _________.
3. Parallel lines have _________ solutions.
4. In the substitution method, we replace one _________ in the other equation.
5. If a₁/a₂ = b₁/b₂ = c₁/c₂, the lines are _________.

✓ B. True or False

1. A pair of parallel lines has one solution. _______
2. The graphical method always gives an exact answer. _______
3. Elimination method removes one variable by adding or subtracting. _______
4. If a₁/a₂ ≠ b₁/b₂, the pair is inconsistent. _______
5. Coincident lines have infinitely many solutions. _______

✏️ C. Solve These

1. Solve by substitution: x + 3y = 11 and 2x − y = 1.

Show your work here

2. Solve by elimination: 5x + 2y = 22 and 3x + 2y = 14.

Show your work here

🔗 D. Match the Columns

Column A	Column B
1. Lines intersect	a. No solution
2. Lines are parallel	b. Infinite solutions
3. Lines are the same	c. Substitution
4. Replace a variable	d. One solution

🎨 E. Draw / Label

On a graph, draw the lines x + y = 5 and x − y = 1. Mark the point where they meet.

Draw your graph here

✅ Detailed Solutions

A. Fill in the Blanks — Answers

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

2 If two lines intersect, the pair is called consistent (with a unique solution).

3 Parallel lines have no solutions.

4 In the substitution method, we replace one variable in the other equation.

5 If a₁/a₂ = b₁/b₂ = c₁/c₂, the lines are coincident.

B. True or False — Answers

1 FALSE — Parallel lines have NO solution (inconsistent pair).

2 FALSE — Graphical method can give approximate answers when lines cross at non-integer points.

3 TRUE — Elimination removes one variable by adding or subtracting equations.

4 FALSE — If a₁/a₂ ≠ b₁/b₂ the lines intersect (consistent), not inconsistent.

5 TRUE — Coincident lines represent the same line; every point is a solution.

C1. Substitution: x + 3y = 11 and 2x − y = 1

1 From equation 1: x = 11 − 3y

2 Substitute in equation 2: 2(11 − 3y) − y = 1

3 22 − 6y − y = 1 → −7y = −21 → y = 3

4 Substitute back: x = 11 − 3(3) = 11 − 9 = x = 2

x = 2, y = 3

C2. Elimination: 5x + 2y = 22 and 3x + 2y = 14

1 Subtract equation 2 from equation 1: (5x − 3x) + (2y − 2y) = 22 − 14

2 2x = 8 → x = 4

3 Substitute x = 4 into equation 2: 3(4) + 2y = 14 → 12 + 2y = 14

4 2y = 2 → y = 1

x = 4, y = 1

D. Match the Columns — Answers

1 Lines intersect → d (One solution)

2 Lines are parallel → a (No solution)

3 Lines are the same → b (Infinite solutions)

4 Replace a variable → c (Substitution method)

E. Lines x + y = 5 and x − y = 1 — Intersection

1 Adding both equations: 2x = 6 → x = 3

2 Substitute: 3 + y = 5 → y = 2

3 The lines meet at (3, 2).

Intersection point: (3, 2)

🃏 Section 6 — Revision Flashcards

👉

📐

What is a pair of linear equations?

Two equations with the same two variables (x and y). We find values that satisfy both.

📈

What does consistent mean?

The pair has at least one solution. The lines either intersect or overlap completely.

🚫

When is a pair inconsistent?

When the lines are parallel. They never meet, so there is no solution.

🔢

What is the ratio test?

Compare a₁/a₂ and b₁/b₂. If not equal = one solution. If equal, check c₁/c₂ for parallel or same.

🔄

How does substitution work?

Write one variable in terms of the other, put it into the second equation, and solve.

❌

How does elimination work?

Make coefficients of one variable equal, then add or subtract to remove it.

♾

What if you get 0 = 0 while solving?

The lines are the same! There are infinitely many solutions (dependent pair).

🔎

What is a graphical solution?

Draw both lines on a graph. The point where they cross is the solution (x, y).

Class 10 Mathematics — Pair of Linear Equations in Two Variables

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📐 Class 10 Mathematics — Pair of Linear Equations in Two Variables Interactive Visual Revision Notes Aligned with the Class 10 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.

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