Solve a system of two linear equations using Cramer's rule
①x +y =
②x +y =
Result
Solution
Step-by-Step Derivation
System Solving Formula
D = A₁B₂ - A₂B₁ Dx = C₁B₂ - C₂B₁, Dy = A₁C₂ - A₂C₁ x = Dx/D, y = Dy/D
Cramer's Rule uses determinants to directly compute the unique solution of a linear system. When D=0, the two lines are parallel or coincident, requiring separate judgment.
⚠Note: When D=0 (lines are parallel or coincident), the system has no unique solution. If the two equations are scalar multiples, they are coincident (infinitely many solutions); otherwise, they are parallel (no solution).
What Is a System of Equations?
Solving a system of two linear equations means finding values of x and y that satisfy both equations simultaneously. Geometrically, this corresponds to finding the intersection point of two lines.
Cramer's Rule
Uses determinants D, Dx, Dy to directly compute x=Dx/D, y=Dy/D. Elegant and standardized.
Substitution Method
Solve one equation for y in terms of x, substitute into the other equation, solve for x, then back-substitute for y.
Elimination Method
Multiply equations by constants to equalize coefficients, then add or subtract to eliminate one variable and solve sequentially.
Geometric Meaning
Every point on a line satisfies its equation. The intersection point satisfies both lines simultaneously, corresponding to the unique solution.
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Frequently Asked Questions
What is Cramer's rule?▼
Cramer's rule uses determinants to solve linear systems. Let D=A₁B₂-A₂B₁, Dₓ=C₁B₂-C₂B₁, Dᵧ=A₁C₂-A₂C₁. Then x=Dₓ/D, y=Dᵧ/D. When D≠0, the system has a unique solution.
How does no solution relate to parallelism?▼
When D=0, both lines have the same slope (they are parallel). If C₁/C₂ also equals A₁/A₂ (the equations are scalar multiples), the lines are coincident with infinitely many solutions. Otherwise, the lines are truly parallel with no intersection, meaning no solution.
How to use the substitution method?▼
From the first equation, solve for y = ... (in terms of x). Substitute this expression into the second equation, eliminating y and leaving a single equation in x. Solve for x, then plug back into y = ... to find y.
How to use the elimination method?▼
Multiply each equation by a constant so that the coefficients of one variable (say, x) have the same absolute value. Then subtract (or add) the two equations to eliminate x, leaving a single equation in y. Solve for y, then back-substitute to find x.
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