01. Linear Equation in Two Variables
A linear equation in two variables x and y is generally written as:
where a, b and c are real numbers and at least one of a and b
is non-zero.
💡 Easy Hindi:
Equation mein x aur y ki highest power 1 honi chahiye.
Isliye ye linear equation hai.
✏ Example
2x + 3y − 7 = 0
Here:
a = 2, b = 3, c = −7
Therefore, it is a linear equation in two variables.
02. Pair of Linear Equations
Two linear equations in the same two variables form a
pair of linear equations in two variables.
✏ Example
2x + 3y − 7 = 0
4x − y − 5 = 0
These two equations form a pair of linear equations.
💡 Simple meaning: Same x-y variables wali do linear
equations ko pair bolte hain.
03. Solution of a Pair
A pair of values (x, y) is called a solution if it satisfies
both equations simultaneously.
✏ Example
Check whether (2, 1) is a solution of:
x + y = 3
2x − y = 3
First equation:
2 + 1 = 3 ✓
Second equation:
2(2) − 1 = 3 ✓
Therefore,
(2, 1) is the solution.
💡 Sirf ek equation satisfy karna enough nahi hai.
Dono equations satisfy honi chahiye.
04. Graphical Meaning of a Pair
Each linear equation in two variables represents a
straight line.
Therefore, a pair of linear equations represents
two straight lines.
Their relative positions determine the number of solutions.
| Position of Lines |
Number of Solutions |
Type |
| Intersecting |
One |
Unique solution |
| Parallel |
None |
No solution |
| Coincident |
Infinitely many |
Infinite solutions |
💡 Graph trick:
Cross/intersect → 1 solution
Parallel → 0 solution
Same line → infinitely many solutions
05. Consistent and Inconsistent Pair ⭐
A pair of linear equations is called consistent if it has
at least one solution.
A pair is called inconsistent if it has no solution.
✏ Example 1 — Consistent
x + y = 5
x − y = 1
These equations have one solution.
Therefore, the pair is
consistent.
✏ Example 2 — Inconsistent
x + y = 4
x + y = 7
Both cannot be true at the same time.
Therefore, there is
no solution.
Hence, the pair is
inconsistent.
💡 Yaad rakho:
At least one solution → Consistent
No solution → Inconsistent
06. Conditions for Number of Solutions ⭐⭐⭐
For:
compare:
a₁/a₂, b₁/b₂, c₁/c₂
| Condition |
Number of Solutions |
Lines |
|
a₁
a₂
≠
b₁
b₂
|
One |
Intersecting |
|
a₁
a₂
=
b₁
b₂
≠
c₁
c₂
|
None |
Parallel |
|
a₁
a₂
=
b₁
b₂
=
c₁
c₂
|
Infinitely many |
Coincident |
💡 Super shortcut:
First two ratios different → one solution
First two equal, third different → no solution
All three equal → infinitely many solutions
07. Example — Unique Solution
✏ Example
Check the number of solutions:
2x + 3y − 5 = 0
4x + y − 7 = 0
Here:
a₁ = 2, b₁ = 3
a₂ = 4, b₂ = 1
a₁
a₂
=
2
4
=
1/2
b₁
b₂
=
3/1 = 3
Since:
1/2 ≠ 3
Unique solution.
💡 Ratios equal nahi hain → lines intersect → one solution.
08. Example — No Solution
✏ Example
3x + 2y − 5 = 0
6x + 4y − 9 = 0
a₁
a₂
=
3
6
= 1/2
b₁
b₂
=
2
4
= 1/2
But:
c₁
c₂
=
−5
−9
≠ 1/2
Therefore:
No solution.
💡 First two ratios same but third different → parallel lines →
no solution.
09. Example — Infinitely Many Solutions
✏ Example
2x + 3y − 5 = 0
4x + 6y − 10 = 0
2
4
=
3
6
=
−5
−10
= 1/2
All three ratios are equal.
Therefore:
Infinitely many solutions.
10. Substitution Method ⭐⭐⭐
In the substitution method, we express one variable in terms
of the other and substitute it into the second equation.
Steps:
1. Take either equation and express one variable in terms of
the other.
2. Substitute this expression in the second equation.
3. Find the value of one variable.
4. Substitute it back to find the other variable.
✏ Example
Solve:
x + y = 7
x − y = 1
From first equation:
x = 7 − y
Substitute in second:
(7 − y) − y = 1
7 − 2y = 1
−2y = −6
y = 3
Now:
x = 7 − 3
x = 4
Therefore, x = 4, y = 3.
💡 Easy Hindi:
Pehle ek variable ko akela karo, phir uski value/expression
dusri equation mein put karo.
11. Substitution Method — Another Type
✏ Example
Solve:
2x + y = 8
x − y = 1
From second equation:
y = x − 1
Substitute in first:
2x + (x − 1) = 8
3x − 1 = 8
3x = 9
x = 3
Therefore:
y = 3 − 1 = 2
x = 3, y = 2
12. Elimination Method ⭐⭐⭐
In the elimination method, we eliminate one variable by adding
or subtracting the equations after making their coefficients
equal in magnitude.
Steps:
1. Arrange both equations properly.
2. Make coefficients of one variable equal or opposite.
3. Add or subtract the equations.
4. Find one variable.
5. Substitute its value to find the other variable.
✏ Example
Solve:
2x + 3y = 13
2x − y = 5
Subtract second equation from first:
(2x + 3y) − (2x − y) = 13 − 5
4y = 8
y = 2
Substitute in:
2x − y = 5
2x − 2 = 5
2x = 7
x = 7/2
Therefore, x = 7/2 and y = 2.
💡 Elimination ka main idea:
ek variable ko cancel karo.
13. Elimination with Different Coefficients
✏ Example
Solve:
2x + 3y = 8
3x + 2y = 7
Multiply first equation by 3:
6x + 9y = 24
Multiply second equation by 2:
6x + 4y = 14
Subtract:
5y = 10
y = 2
Put y = 2 in:
3x + 2(2) = 7
3x + 4 = 7
x = 1
x = 1, y = 2
💡 Coefficients same nahi hain → suitable multiplication karke
same coefficient banao.
14. Choosing the Better Method
| Situation |
Useful Method |
| One variable easily isolated |
Substitution |
| Coefficients easily matched |
Elimination |
| Question asks graphically |
Graphical Method |
| Question asks nature of solutions |
Ratio Conditions |
💡 Method choose karna bhi exam skill hai.
Jo method calculation ko shortest banaye, use prefer karo.
15. Situational Problems — General Strategy ⭐⭐⭐
Step 1: Define the variables.
Step 2: Translate the information into two equations.
Step 3: Solve the pair.
Step 4: Interpret the answer according to the question.
💡 Hindi trick:
Pehle “x kya hai?” aur “y kya hai?” clearly define karo.
Uske baad statement ko equations mein convert karo.
16. Word Problem — Numbers
✏ Example
The sum of two numbers is 35 and their difference is 7.
Find the numbers.
Let the numbers be x and y.
According to question:
x + y = 35
x − y = 7
Add both equations:
2x = 42
x = 21
Then:
21 + y = 35
y = 14
Therefore, the numbers are 21 and 14.
17. Word Problem — Cost
✏ Example
The cost of 2 pens and 3 pencils is ₹34.
The cost of 3 pens and 2 pencils is ₹31.
Find the cost of each.
Let:
Cost of one pen = ₹x
Cost of one pencil = ₹y
Equations:
2x + 3y = 34
3x + 2y = 31
Multiply first by 3:
6x + 9y = 102
Multiply second by 2:
6x + 4y = 62
Subtract:
5y = 40
y = 8
Put y = 8:
3x + 16 = 31
3x = 15
x = 5
Pen = ₹5, Pencil = ₹8
18. Word Problem — Age
✏ Example
The sum of the ages of a father and son is 50 years.
The father's age is 4 times the son's age.
Find their ages.
Let son's age = x years.
Father's age = y years.
Equation 1:
x + y = 50
Equation 2:
y = 4x
Substitute:
x + 4x = 50
5x = 50
x = 10
y = 40
Son = 10 years, Father = 40 years
19. Word Problem — Speed and Distance
For such questions, remember:
The given conditions are converted into equations.
✏ Example
A car travels 120 km partly at 40 km/h and partly at
60 km/h. If the total travel time is 2.5 hours, find the
distance travelled at each speed.
Let distance at 40 km/h = x km.
Distance at 60 km/h = y km.
Total distance:
x + y = 120
Time:
x
40
+
y
60
= 2.5
This can be simplified and solved as a pair of linear equations.
💡 Word problem mein formula directly use karke equations banana
sabse important step hai.
20. Exam Type — Find the Value of k
✏ Example
Find k for which the pair has infinitely many solutions:
2x + 3y − 5 = 0
4x + 6y + k = 0
For infinitely many solutions:
a₁
a₂
=
b₁
b₂
=
c₁
c₂
We have:
2/4 = 3/6 = 1/2
Therefore:
−5/k = 1/2
k = −10
k = −10
21. Exam Type — Find k for No Solution
✏ Example
Find k so that:
3x + 2y − 5 = 0
6x + ky − 9 = 0
has no solution.
For no solution:
a₁
a₂
=
b₁
b₂
≠
c₁
c₂
So:
3/6 = 2/k
1/2 = 2/k
k = 4
Check:
−5/−9 = 5/9
and 1/2 ≠ 5/9.
Therefore, k = 4.
22. Common Mistakes ⚠️
1. x + a ko x = a mat lena.
Correct: x + a = x − (−a).
2. Ratio conditions mein signs carefully handle karo.
3. Unique, no solution and infinite solutions ko mix mat karo.
4. Word problems mein variables define kiye bina equations
mat banana.
5. Final answer ko question ke context mein interpret karo.
6. Elimination mein equation ko multiply karte waqt
RHS ko bhi multiply karo.
7. Substitution mein brackets carefully open karo.
23. Formula & Condition Sheet ⭐⭐⭐
24. Quick Revision — Chapter 3
Two linear equations → Pair of linear equations.
Each equation represents a straight line.
Intersecting lines → one solution.
Parallel lines → no solution.
Coincident lines → infinitely many solutions.
At least one solution → consistent.
No solution → inconsistent.
Substitution → express one variable and substitute.
Elimination → cancel one variable.
Word problem → define variables → form equations → solve → interpret.
25. THE GOAL Exam Strategy 🎯
Question says “number of solutions”?
→ Compare the three coefficient ratios.
Question says “solve”?
→ Use substitution or elimination.
Question says “graphically”?
→ Think of the two straight lines and their intersection.
Question says “consistent”?
→ Check whether at least one solution exists.
Question gives a real-life situation?
→ Define x and y first, then form two equations.
💡 Final Hindi Tip:
Question ko dekhte hi method identify karna seekho.
Aadha question wahi solve ho jata hai jab tumhe pata ho
ki kaunsi condition/method use karni hai.