Elimination Method
Solve systems of two linear equations using the elimination method. Enter coefficients of ax+by=e and cx+dy=f to find x and y with Cramer's rule and step-by-step work.
About This Calculator
The Elimination Method Calculator solves systems of two linear equations using Cramer's rule, which is the algebraic foundation of the elimination technique. For the system ax + by = e and cx + dy = f, the calculator computes the determinant and uses it to find the unique solution when one exists.
The elimination method works by adding or subtracting multiples of the equations to cancel one variable. Cramer's rule provides a direct formula: the determinant |a b; c d| = ad - bc. If the determinant is nonzero, x = (ed - bf)/(ad - bc) and y = (af - ec)/(ad - bc). A zero determinant means the lines are parallel (no solution) or coincident (infinite solutions).
This tool is ideal for algebra students learning systems of equations, professionals solving small linear systems, and anyone who needs quick verification of their manual elimination work. The step-by-step output shows the determinant calculation and the values of x and y.
Frequently Asked Questions
How does the elimination method work?
The elimination method solves systems of linear equations by adding or subtracting the equations to cancel one variable. Multiply one or both equations by constants so that the coefficients of one variable are opposites, then add to eliminate that variable and solve for the other.
When should you use elimination instead of substitution?
Use elimination when the coefficients align easily for cancellation, especially with integer coefficients. Elimination is often faster than substitution when both equations are in standard form.
What if the elimination method gives 0 = 0?
If you get a true statement like 0 = 0, the system has infinitely many solutions (coincident lines). This means the two equations represent the same line.
What if elimination gives a false statement?
If you get a false statement like 0 = 5, the system has no solution (parallel lines). This means the two lines never intersect.
What is Cramer's rule?
Cramer's rule is a formula that solves linear systems using determinants. For ax+by=e and cx+dy=f, the determinant is ad-bc. If nonzero, x = (ed-bf)/(ad-bc) and y = (af-ec)/(ad-bc).
Can elimination handle three variables?
Yes, elimination extends naturally to three or more variables. You eliminate one variable at a time by combining equations, then back-substitute to find all solutions.
What does the determinant tell you about a system?
A zero determinant means the equations are dependent (parallel or coincident), so there is either no solution or infinitely many. A nonzero determinant guarantees a unique solution.
Is this calculator free?
Yes, all calculators on Calculy are completely free to use with no limits or registration required.