Class 7 | Ganita Prakash Part-II | Chapter 7
🔍 Finding the Unknown
Linear Equations · Balance · Solve · History of Algebra
⚖️ 7.1 — Find the Unknowns
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Think of equations as a balance scale! ⚖️
Both sides must be equal (balanced).
Example: 2 + 2 = 4 ✅ (balanced!)
If 4 + x = 7 → what is x?
Remove 4 from both sides → x = 3 ✅
Both sides must be equal (balanced).
Example: 2 + 2 = 4 ✅ (balanced!)
If 4 + x = 7 → what is x?
Remove 4 from both sides → x = 3 ✅
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The Golden Rule of Equations:
Whatever you do to one side, DO THE SAME to the other side!
Add? Both sides. Subtract? Both sides. Multiply? Both sides. Divide? Both sides.
Whatever you do to one side, DO THE SAME to the other side!
Add? Both sides. Subtract? Both sides. Multiply? Both sides. Divide? Both sides.
Removing equal weights from both sides:
If sacks + 3 = sacks + sacks, remove sacks from both:
3 = one sack → each sack weighs 3! ✅
(This is exactly what we do in algebra!)
If sacks + 3 = sacks + sacks, remove sacks from both:
3 = one sack → each sack weighs 3! ✅
(This is exactly what we do in algebra!)
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📐 Matchstick & Pattern Equations
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Matchstick patterns lead to equations!
1 square = 4 sticks
2 squares = 7 sticks (4 + 3)
3 squares = 10 sticks (4 + 3 + 3)
n squares = 3n + 1 sticks
If someone used 25 sticks → 3n + 1 = 25 → n = 8 squares!
1 square = 4 sticks
2 squares = 7 sticks (4 + 3)
3 squares = 10 sticks (4 + 3 + 3)
n squares = 3n + 1 sticks
If someone used 25 sticks → 3n + 1 = 25 → n = 8 squares!
🎭 Real life equations everywhere!
Age problems, money problems, distance problems — all become equations!
"Rima is 5 years older than Mia. Their ages add to 25. How old is each?"
Let Mia = x, Rima = x + 5. x + (x+5) = 25 → 2x = 20 → x = 10!
Age problems, money problems, distance problems — all become equations!
"Rima is 5 years older than Mia. Their ages add to 25. How old is each?"
Let Mia = x, Rima = x + 5. x + (x+5) = 25 → 2x = 20 → x = 10!
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🔣 7.2 — Solving Equations Systematically
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Steps to Solve a Linear Equation:
Move all variable terms to one side, constants to the other
Simplify: combine like terms on each side
Divide both sides by the coefficient of the variable
Check: substitute back to verify!
Examples:
3x + 5 = 20
3x = 20 – 5 = 15
x = 15 ÷ 3 = 5
Check: 3(5) + 5 = 15 + 5 = 20 ✅
2x – 3 = x + 4
2x – x = 4 + 3
x = 7
Check: 2(7)–3 = 11, 7+4 = 11 ✅
3x + 5 = 20
3x = 20 – 5 = 15
x = 15 ÷ 3 = 5
Check: 3(5) + 5 = 15 + 5 = 20 ✅
2x – 3 = x + 4
2x – x = 4 + 3
x = 7
Check: 2(7)–3 = 11, 7+4 = 11 ✅
Transposing (Moving across the = sign):
When a term moves to the other side, its sign CHANGES!
+5 moves → becomes –5
–3 moves → becomes +3
×2 moves → becomes ÷2
÷4 moves → becomes ×4
When a term moves to the other side, its sign CHANGES!
+5 moves → becomes –5
–3 moves → becomes +3
×2 moves → becomes ÷2
÷4 moves → becomes ×4
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📜 7.4 — A Pinch of History
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🏛️ Ancient Algebra!
Brahmagupta (628 CE, India) wrote about solving equations!
Al-Khwarizmi (820 CE, Baghdad) wrote "Al-Kitab al-mukhtasar fi hisab al-jabr" — this is where the word "ALGEBRA" comes from! (al-jabr = reunion of broken parts)
Diophantus (250 CE, Alexandria) used symbols for unknowns.
Brahmagupta (628 CE, India) wrote about solving equations!
Al-Khwarizmi (820 CE, Baghdad) wrote "Al-Kitab al-mukhtasar fi hisab al-jabr" — this is where the word "ALGEBRA" comes from! (al-jabr = reunion of broken parts)
Diophantus (250 CE, Alexandria) used symbols for unknowns.
The word "Algorithm" also comes from Al-Khwarizmi's name!
Medieval Europeans heard "Al-Khwarizmi" and it became "algoritmi" in Latin, then "algorithm"!
One mathematician's name gave us two important words! 🎉
Medieval Europeans heard "Al-Khwarizmi" and it became "algoritmi" in Latin, then "algorithm"!
One mathematician's name gave us two important words! 🎉
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📝 Quick Cheatsheet
Balance both sides!
What you do to L
do to R too ⚖️
What you do to L
do to R too ⚖️
Transpose: move
across = sign →
sign changes!
across = sign →
sign changes!
3x+5=20
→ 3x=15
→ x=5 ✅
→ 3x=15
→ x=5 ✅
Always CHECK:
Substitute x back
Both sides equal?
Substitute x back
Both sides equal?
| Type | Form | Solution |
|---|---|---|
| Simple | x + a = b | x = b – a |
| With coefficient | ax = b | x = b ÷ a |
| Two-step | ax + b = c | x = (c – b) ÷ a |
| Both sides | ax + b = cx + d | Move all x to one side, numbers to other |
🌟 Chapter 7 Complete! 🌟
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