Rearrange linear algebraic equations
Scenario
Aap market gaye aur aam ka rate `₹60` per kg tha. Aapne kuch kgs aam liye aur total `₹180` diye. Aapko total pata hai aur rate pata hai, toh aap apne aap hisaab laga lete ho ki aapne `3 kg` aam liye.
Equations mein unknown cheez ko baaki sab ke terms mein kaise isolate karte hain?
Mental model
Equation ek weighing scale (tarazu) jaisa hota hai, agar left mein kuch karo toh right mein bhi same karna padta hai.
Rearranging equation ka main goal us variable ko akela (isolate) karna hai jiski value aapko dhoondhni hai. Agar koi number add ho raha hai, toh use doosri side le jaa kar subtract kardo. Agar multiply ho raha hai, toh udhar jaake divide kardo. Inverse operations use hote hain.
Explanation
Physics ke formulas ek rigid structure follow karte hain, lekin numerical solve karte waqt aapko hamesha same cheez nikalne ko nahi milti. Jaise `v = u + at` mein, kabhi aapko `v` dhoondhna hoga jahan sab values sidhi right side par di hoti hain. Par kabhi aapko `u` ya `t` dhoondhna padta hai, tab equation ko rearrange karna zaroori ho jata hai.
Equation ko rearrange karne ka sabse basic asool (rule) yeh hai ki 'equal to' (`=`) ke sign ke dono taraf balance maintain hona chahiye. Jisko dhoondhna hai (unknown variable), usko ek side par akela (isolate) chhodna hai. Iske liye hum inverse operations (ulati kriya) ka use karte hain.
Agar variable ke saath koi term `+` ho rahi hai, toh usko equality ke doosri taraf bhejne par woh `-` ban jayegi (aur ulta bhi). Agar variable ke saath koi number multiply ho raha hai, toh woh doosri side jaa kar denominator (divide) mein chala jayega.
Example ke liye, `a = (v - u) / t` mein se `t` ko isolate karna hai, toh pehle `t` jo divide mein hai usko left side bhejenge multiply hone (`a * t = v - u`). Phir `a` jo multiply ho raha hai, usko right side bhejo jahan woh divide hoga: `t = (v - u) / a`. Ek baar variable isolate ho jaye, phir aap easily numbers daalkar calculation kar sakte hain. Formula pehle rearrange karna, phir values daalna, galti hone ke chances bohot kam kar deta hai.
Worked example
Problem: Rearrange the formula `v = u + at` to solve for the time `t`.
- Pehle `u` ko left side bhejkar subtract karenge taaki `at` akela bach jaye.
- Ab `t` ke saath `a` multiply ho raha hai, toh usko left side bhejkar divide karenge.
Answer: `t = (v - u) / a`
Problem: Given the formula `s = d / t`, rearrange it to find distance `d`.
- Yahan `d` ko isolate karna hai. Uske niche `t` divide ho raha hai.
- `t` ko equality ke doosri taraf bhejte hain, toh yeh multiply ho jayega.
Answer: `d = s * t`
Key points
- Inverse operations: `+` becomes `-`, `-` becomes `+`, `*` becomes `/`, and `/` becomes `*` when moved across the equals sign.
- Isolate the unknown: Your main goal is to leave the variable you want to find all by itself on one side of the equation.
Common mistakes
- Moving terms with the wrong sign: Addition (`+`) wale term ko doosri taraf bhejne par kai baar bache use dobara add hi kar dete hain, jabki wo minus (`-`) hona chahiye.
- Breaking multiplication pairs too early: Equation `3x + 4 = 10` mein pehle `+4` ko udhar bhejna chahiye (`10 - 4`). Kuch students pehle hi `3` ko divide karne lagte hain, jisse fractions me mess create ho jata hai.
Recall questions
- What is the inverse operation of multiplication?
- If x + 5 = 12, what is the value of x?
- Rearrange F = m * a to solve for m.
Questions & answers
If an equation is given as 2x - 3 = 9, what is the value of x?
6
Approach: Move -3 to the right side to become +3 (2x = 12). Then divide by 2 (x = 6).
Solve for a in the equation: v = u + at, given v=20, u=0, t=5.
4
Approach: First rearrange to a = (v - u) / t. Then substitute values: a = (20 - 0) / 5 = 20 / 5 = 4.
Continue learning
Next: Apply v = u + at