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Systems of Equations

2×2: ax+by=e, cx+dy=f. D=ad−bc. D≠0: unique solution. D=0: no solution or infinitely many. Elimination: add multiples to cancel variable. Gaussian: row reduction for 3×3.

Concept Fundamentals
D=ad−bc
2×2
D≠0
Unique
Add to cancel
Elimination
Row reduction
Gaussian

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D=0: lines parallel (inconsistent) or same (dependent). Elimination: add k×eq1 to eq2 to cancel a variable. Choose k so coefficients match. Gaussian: pivot, eliminate, back-substitute. Same idea as elimination, organized.

Key quantities
D=ad−bc
2×2
Key relation
D≠0
Unique
Key relation
Add to cancel
Elimination
Key relation
Row reduction
Gaussian
Key relation

Ready to run the numbers?

Why: Systems model multiple constraints — supply/demand, mixtures, circuits. Determinant D reveals solution type. Elimination is systematic. Gaussian extends to any size. Used in engineering and economics.

How: 2×2: compute D=ad−bc. If D≠0, use Cramer or elimination. Elimination: multiply equations to match coefficients, add to eliminate. 3×3: Gaussian elimination — row reduction to upper triangular.

D=0: lines parallel (inconsistent) or same (dependent).Elimination: add k×eq1 to eq2 to cancel a variable. Choose k so coefficients match.
Sources:Khan Academy — Systems of EquationsWolfram MathWorld — Linear System

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Solve Systems2×2, 3×3, elimination

📐 Examples — Click to Load

Coefficients

systems.sh
CALCULATED
$ solve --system
Solution
x=3.0000, y=1.0000
Classification
Consistent independent
Determinant
-3
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Solution Components

Method Efficiency

📐 Steps (elimination)

Equation 1: 2x + 1y = 7
Equation 2: 1x + -1y = 2
Determinant D = 2×-1 - 1×1 = -3
Multiply Eq1 by -1, Eq2 by 1 to align y coefficients
-2x + -1y = -7
1x + -1y = 2
Subtract: -3x = -9
x = 3.0000
Substitute: y = (7 - 2×3.0000) / 1 = 1.0000
Solution: (3.0000, 1.0000)

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

2x+y=5, x−y=1. Add: 3x=6, x=2. Then y=1.

— Elimination

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D=0 means coefficient matrix singular — no unique solution.

— Determinant

📋 Key Takeaways

  • • Elimination: Add/subtract equations to cancel a variable
  • • Gaussian elimination: Row operations to reach row echelon form
  • • Classification: Consistent independent (one), dependent (infinite), inconsistent (none)
  • • Determinant: For 2×2, det=0 means no unique solution
  • • Applications: Word problems, ages, coins, mixture, distance-rate-time

📖 Elimination Method

  1. Multiply equations so one variable has opposite coefficients
  2. Add or subtract to eliminate that variable
  3. Solve the resulting single-variable equation
  4. Back-substitute to find the other variable(s)

📖 Gaussian Elimination (3×3)

Use row operations to get zeros below the diagonal (row echelon form), then back-substitute.

❓ FAQ

When is there no solution?

When the lines/planes are parallel (same normal, different constant). The system is inconsistent.

When are there infinite solutions?

When equations are multiples of each other (dependent). The solution set is a line or plane.

What is row echelon form?

A matrix form where each row has more leading zeros than the previous. Enables back-substitution.

How does the determinant help?

For 2×2, if det=0 the system has no unique solution (either none or infinitely many).

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