Chapter 12 — Linear Equations in Two Variables

Section 12.1

Linear Equations — Graphical Method

A linear equation in two variables ax + by + c = 0 represents a straight line. Its solution is any point (x,y) on that line.

📐 Standard Forms

ax + by + c = 0 or y = mx + b

Slope-intercept form: y = mx + b (m = slope, b = y-intercept)
A single linear equation has infinitely many solutions — each point on the line.

Two lines y=2x-1 and y=-x+6 intersect at one point → unique solution (consistent)

Section 12.2 — NEW for Class 9 (from Cl.10)

Pair of Linear Equations

A pair of linear equations in two variables: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. The solution is the point(s) satisfying BOTH equations simultaneously.

📐 Consistency Conditions

Compare ratios a₁/a₂, b₁/b₂, c₁/c₂:
• a₁/a₂ ≠ b₁/b₂ → Unique solution (lines intersect) → Consistent
• a₁/a₂ = b₁/b₂ = c₁/c₂ → Infinite solutions (lines coincide) → Consistent (dependent)
• a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → No solution (lines parallel) → Inconsistent

Type	Condition	Graph	Solutions
Consistent	a₁/a₂ ≠ b₁/b₂	Lines intersect at 1 point	Unique (1 solution)
Consistent (Dependent)	a₁/a₂ = b₁/b₂ = c₁/c₂	Lines coincide (same line)	Infinitely many
Inconsistent	a₁/a₂ = b₁/b₂ ≠ c₁/c₂	Lines are parallel	No solution

Section 12.3 — NEW Method for Class 9

Substitution Method ★

Express one variable in terms of the other from one equation, then substitute into the second. Best when one equation is easy to isolate a variable.

Example 1 · 3M

Solve: x + y = 14 and x – y = 4. [Substitution]

1

From eq (1): x = 14 – y …(*)

2

Substitute (*) into eq (2): (14–y) – y = 4 → 14–2y=4 → y=5

3

x = 14–5 = x=9

✓

Check: 9+5=14 ✓, 9–5=4 ✓

x = 9, y = 5

Example 2 · 3M

Solve: 3x + 2y = 11 and 2x – y = 0.

1

From eq (2): y = 2x

2

Sub into (1): 3x+2(2x)=11 → 7x=11 → x=11/7

3

y = 2×11/7 = 22/7

x = 11/7, y = 22/7

Section 12.4 — NEW Method for Class 9

Elimination Method ★

Multiply equations to make coefficients of one variable equal, then add or subtract to eliminate that variable. Best for messy fractions.

Example 3 · 3M

Solve: 2x + 3y = 13 and 5x – 4y = –2. [Elimination]

1

Multiply (1)×4: 8x+12y=52 Multiply (2)×3: 15x–12y=–6

2

Add: 23x=46 → x=2

3

Sub x=2 in (1): 4+3y=13 → y=3

x = 2, y = 3

Example 4 · 5M Word Problem

Sum of two numbers is 30. If the larger exceeds twice the smaller by 3, find them.

1

Let larger = x, smaller = y. Equations: x+y=30 and x=2y+3

2

Sub x=2y+3 into x+y=30: 3y+3=30 → y=9

3

x=2(9)+3=21. Check: 21+9=30 ✓, 21=2(9)+3 ✓

The two numbers are 21 and 9.

Section 12.5

Consistency & Graphical Interpretation

Example 5 · 2M

Check if 2x+3y=5 and 4x+6y=9 are consistent.

1

a₁/a₂=2/4=1/2, b₁/b₂=3/6=1/2, c₁/c₂=5/9

2

a₁/a₂=b₁/b₂≠c₁/c₂ → Inconsistent (parallel lines, no solution)

Inconsistent — no solution. Lines are parallel.

More Examples

Word Problems

Example 6 · 5M

A train travels 360km. If speed increases by 5km/h it takes 1hr less. Find the original speed.

1

Let speed = v km/h. Time = 360/v. At (v+5): 360/(v+5) = 360/v – 1

2

360v – 360(v+5) = –v(v+5) → –1800 = –v²–5v → v²+5v–1800=0

3

(v+45)(v–40)=0 → v=40 km/h (taking positive value)

Original speed = 40 km/h

Example 7 · 3M

Age problem: Father is 42yrs. He is 6 times his daughter's age. When will he be twice her age?

1

Father=42, Daughter=7. Let after x years: 42+x=2(7+x)

2

42+x=14+2x → x=28

After 28 years — father (70) will be twice daughter's age (35).

NCERT Exercise · Solved

Exercise 12.1 & 12.2

Q1 · 2M

Solve graphically: x + y = 6 and x – y = 2.

1

x+y=6: (0,6),(6,0),(3,3). x–y=2: (0,–2),(2,0),(4,2)

2

Intersection: add equations → 2x=8 → x=4, y=2. Point: (4,2)

x = 4, y = 2

Q2 · 3M

Solve by elimination: 3x + 4y = 10 and 2x – 2y = 2.

1

(1)×2: 6x+8y=20. (2)×3: 6x–6y=6. Subtract: 14y=14 → y=1

2

3x+4=10 → x=2

x = 2, y = 1

Quick Reference

Formula & Method Summary

Concept	Key Formula / Rule
Slope-intercept form	y = mx + b (m=slope, b=y-intercept)
General form	ax + by + c = 0
Unique solution	a₁/a₂ ≠ b₁/b₂ (lines intersect)
No solution	a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (parallel)
∞ solutions	a₁/a₂ = b₁/b₂ = c₁/c₂ (coincident)
Substitution	Express one variable, substitute in other equation
Elimination	Make one coefficient equal, add/subtract to eliminate

Smart Study

8 Tips — Linear Equations

1

Always check your answer

After solving, substitute x and y back into BOTH original equations. Both must be satisfied. CBSE gives marks for the check.

2

Choose the easier variable to isolate

In substitution, pick the variable with coefficient 1 (no division needed). In elimination, pick the variable whose coefficients share an LCM quickly.

3

Consistency: use ratio test

For 2×2 system: compute a₁/a₂, b₁/b₂, c₁/c₂. Three equal ratios = dependent. Two equal (not third) = inconsistent. Not equal first two = unique solution.

4

Word problems: define variables clearly

"Let x = ..." — always state what each variable represents. Then form two equations from the two given conditions.

5

Graphical method: plot 3 points

For each line, find x-intercept (y=0), y-intercept (x=0), and one more point. All three collinear = correct line drawn.

6

Elimination: multiply carefully

When multiplying an equation, multiply EVERY term including the constant. Missing the constant is the most common error.

7

Add or subtract in elimination?

If the equal coefficients have the SAME sign → subtract the equations. If OPPOSITE signs → add. This eliminates the variable.

8

Speed / distance / age formula setup

Speed: d=vt. Age: current and future ages form a system. Coins/mixtures: total count and total value form a system.

CBSE Practice

50 Practice Questions

MCQ · 1M · 2M · 3M · 5M · Case-Based

Linear Equationsin Two Variables