Noesis
2.1.9Algebra — Introduction & Intermediate

Linear equations in two variables — graphical and algebraic solutions

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🇮🇳 Yeh note Hinglish mein padho →

1. What is a linear equation in two variables?

WHY a straight line? Solve for yy: y=−abx−cby = -\frac{a}{b}x - \frac{c}{b}. This is y=mx+ky = mx + k — constant slope mm, so equal steps in xx give equal steps in yy. Constant rate of change ⇒ a line.

2. A system of two equations

a1x+b1y=c1a2x+b2y=c2\begin{aligned} a_1x + b_1y &= c_1 \\ a_2x + b_2y &= c_2 \end{aligned}

Three geometric possibilities:

Figure — Linear equations in two variables — graphical and algebraic solutions

WHY these ratios? Slope of line 1 is −a1/b1-a_1/b_1, of line 2 is −a2/b2-a_2/b_2. Equal slopes ⇔ a1a2=b1b2\frac{a_1}{a_2}=\frac{b_1}{b_2} ⇔ parallel or same. If additionally the constant ratio matches, one equation is just a multiple of the other → same line. If not, they are shifted apart → parallel, no solution.

3. Algebraic methods (derived from scratch)

3a. Substitution

3b. Elimination

4. Worked examples

5. Forecast-then-Verify

Recall Before solving

4x+6y=204x+6y=20 and 2x+3y=102x+3y=10, forecast the type. Ratios 42=63=2010=2\frac{4}{2}=\frac{6}{3}=\frac{20}{10}=2. All equal ⇒ same line, infinitely many solutions. Verify: dividing first by 2 gives the second exactly. ✓ Forecast confirmed.

6. Common mistakes

7. Feynman & memory

Recall Explain to a 12-year-old

Imagine two straight roads drawn on a map. One equation = one road. Finding the answer means finding the exact spot where the two roads cross — that's the (x,y)(x,y) that works for both. If the roads are the same road, they touch everywhere (infinite answers). If they're parallel like train tracks, they never touch (no answer).

8. Active recall

General form of a linear equation in two variables?
ax+by+c=0ax+by+c=0 with (a,b)≠(0,0)(a,b)\neq(0,0); graph is a straight line.
How many solutions does ONE linear equation in two variables have?
Infinitely many (every point on its line).
Ratio condition for a UNIQUE solution?
a1a2≠b1b2\frac{a_1}{a_2}\neq\frac{b_1}{b_2}.
Ratio condition for NO solution (parallel)?
a1a2=b1b2≠c1c2\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}.
Ratio condition for INFINITE solutions (same line)?
a1a2=b1b2=c1c2\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}.
Formula for xx by elimination (Cramer)?
x=c1b2−c2b1a1b2−a2b1x=\dfrac{c_1b_2-c_2b_1}{a_1b_2-a_2b_1}.
What does the denominator D=a1b2−a2b1=0D=a_1b_2-a_2b_1=0 mean geometrically?
Lines are parallel or same → no unique solution.
Downstream vs upstream speeds in terms of boat bb & stream ss?
Downstream =b+s=b+s, upstream =b−s=b-s.
Why is the graph a straight line?
y=−abx−cby=-\frac{a}{b}x-\frac{c}{b} has constant slope → equal xx-steps give equal yy-steps.
Best method when one coefficient is 11?
Substitution (isolate that variable cleanly).

Connections

  • Slope-intercept form y = mx + c
  • Simultaneous equations by matrices & determinants
  • Consistency and rank of linear systems
  • Word problems — age, boat-stream, mixture
  • Graphing lines and intercepts
  • Linear inequalities in two variables

Concept Map

graph is

solve for y

explains why

one eqn has

need second eqn

solved by

solved by

method 1

method 2

derives

classified by

D=0 means

D=0 blocks

Linear eqn ax+by+c=0

Straight line

y=mx+k constant slope

Infinitely many solutions

System of two equations

Graphical: find intersection

Algebraic methods

Substitution

Elimination

Cramer formula with D=a1b2-a2b1

Ratio consistency test

Parallel or same line

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, ek linear equation like 2x+3y=122x+3y=12 ka matlab hai ek seedhi line — is line ke har point pe koi na koi (x,y)(x,y) solution milega, isliye ek equation ke infinite solutions hote hain. Ek hi answer chahiye to hume do equations chahiye, aur solution wahi point hai jahan dono lines aapas mein cross karti hain.

Do line kaise mil sakti hain, teen tareeke se: (1) ek baar cross karein → unique solution, (2) bilkul same line ho → infinite solutions, (3) parallel ho jaayein (train ki patri jaisi) → no solution. Kaunsa case hai ye ratios se pata chalta hai: agar a1a2≠b1b2\frac{a_1}{a_2}\neq\frac{b_1}{b_2} to unique; agar teeno ratio barabar to same line; agar sirf pehle do barabar par teesra alag to parallel.

Solve karne ke do algebraic tareeke hain. Substitution: ek equation se ek variable alag karo aur doosri mein daal do — problem chhoti ho jaati hai. Elimination: dono equations ko multiply karke ek variable ka coefficient same banao, phir add/subtract karke usko khatam karo. Jab ek coefficient 11 ho to substitution best, jab coefficients neat ho to elimination fast.

Sabse badi galti: term ko doosri taraf le jaate waqt sign flip karna bhoolna, aur answer ko sirf ek equation mein check karna. Hamesha (x,y)(x,y) ko dono equations mein daal ke verify karo — tabhi wo sach mein intersection point hai. Boat-stream jaise word problems mein yaad rakho: downstream =b+s=b+s, upstream =b−s=b-s. Bas!

Go deeper — visual, from zero

D1FoundationsEvery symbol from zero · 1,802 wordsD2Visual walkthroughThe derivation in pictures · 1,908 wordsD3Worked examplesEvery scenario covered · 2,702 wordsD4ExercisesGraded, full solutions · 2,741 wordsD5Question bankConcept traps, quick reveals · 1,159 words

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Connections

Parallel and perpendicular lines — properties, transversal, alternate - co-interior anglesMaths · 1.2.4Linear equations in one variable — solving, transposition methodMaths · 2.1.7Simultaneous equations — substitution, elimination, cross-multiplicationMaths · 2.1.10