Simultaneous Equations
Solve ax+by=e, cx+dy=f. Cramer: x=(ed−bf)/D, y=(af−ec)/D where D=ad−bc. Elimination: add multiples to cancel variable. Substitution: solve one, plug into other. Matrix: X=A⁻¹B.
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D=0: no unique solution — either no solution (inconsistent) or infinitely many (dependent). Cramer's rule extends to 3×3 using 3×3 determinants. Matrix form AX=B: solution X=A⁻¹B when A is invertible.
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Why: Simultaneous equations model constraints — supply/demand, mixtures, geometry. Cramer gives formula when D≠0. Elimination and substitution work for any consistent system. Used in economics and engineering.
How: Cramer: compute D=ad−bc. If D≠0, x=(ed−bf)/D, y=(af−ec)/D. Elimination: multiply equations to match coefficients, add to eliminate. Substitution: solve one for x or y, substitute.
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📐 Examples — Click to Load
ax + by = e
cx + dy = f
Solution Values
System Classification
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
2x+3y=7, 4x−y=5. D=2(−1)−3(4)=−14. x=(7(−1)−3(5))/−14=2.
— Cramer
D=0 means lines parallel (no solution) or same line (infinite).
— Geometry
📋 Elimination Method
Add or subtract multiples of equations to eliminate one variable. For 2×2: multiply to match coefficients, then add/subtract to get one equation in one unknown.
🔄 Substitution Method
Solve one equation for one variable, substitute into the others. Repeat until one equation in one unknown remains.
📐 Graphical Interpretation
Each equation is a line (2×2) or plane (3×3). Consistent: lines intersect at one point. Inconsistent: parallel lines, no intersection. Dependent: same line, infinitely many solutions.
🔢 Cramer's Rule
For AX = B, x_i = det(A_i)/det(A) where A_i is A with column i replaced by B. Works when det(A) ≠ 0.
📊 Matrix Method
X = A⁻¹B. Compute inverse of coefficient matrix and multiply by constant vector. Requires det(A) ≠ 0.
❓ FAQ
When does a 2×2 system have no solution?
When the lines are parallel: ad - bc = 0 but the system is inconsistent (e.g., 2x+4y=6 and 2x+4y=10).
What is Cramer's rule?
x = Dx/D, y = Dy/D where D is the determinant of the coefficient matrix, Dx and Dy replace the x or y column with the constants.
How many solutions can a linear system have?
Exactly one (consistent), none (inconsistent), or infinitely many (dependent).
What does dependent mean?
The equations represent the same line/plane — every point on it satisfies all equations.
When to use elimination vs substitution?
Elimination is often easier when coefficients are similar. Substitution when one equation is already solved for a variable.
📝 Worked Examples
⚠️ Common Mistakes
- Forgetting to multiply both sides when eliminating — multiply the entire equation.
- Sign errors when subtracting equations — distribute the negative carefully.
- Using Cramer when D=0 — the formula gives division by zero; system is dependent or inconsistent.
📌 Summary
Solve ax+by=e, cx+dy=f using elimination, substitution, or Cramer's rule. Determinant D=ad-bc: if D≠0, unique solution; if D=0, check for inconsistent (no solution) or dependent (∞ solutions). For 3×3, use Cramer or Gaussian elimination. Applications: mixture problems, investment allocation, equilibrium.
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