Elimination
Elimination adds or subtracts equations so that one variable cancels. It is often the fastest method when no variable is easy to isolate.
The idea
If two equations are true, their sum is also true. Adding \(x + y = 10\) and \(x - y = 2\) cancels y: \(2x = 12\), so \(x = 6\), and then \(y = 4\). Elimination works by arranging for one variable's coefficients to be opposites.
Multiplying to match
Often you must multiply first. For \(2x + 3y = 12\) and \(5x - 2y = 11\), multiply the first by 2 and the second by 3: \(4x + 6y = 24\) and \(15x - 6y = 33\). Adding gives \(19x = 57\), so \(x = 3\), and then \(2(3) + 3y = 12\) gives \(y = 2\).
Subtracting equations
If coefficients are equal rather than opposite, subtract. From \(3x + 2y = 16\) and \(3x - y = 7\), subtracting the second from the first gives \(3y = 9\), so \(y = 3\). Then \(3x - 3 = 7\) gives \(x = \tfrac{10}{3}\). Check in the first equation: \(3(\tfrac{10}{3}) + 2(3) = 10 + 6 = 16\). Fractional answers like this are common, which is why elimination beats reading answers off a graph.
Choosing a method
Use substitution when a variable is already isolated or has coefficient 1. Use elimination when the equations are in standard form with awkward coefficients. Both always give the same answer; choose whichever needs less arithmetic.
Special cases with elimination
If both variables cancel, read the result. From \(2x - 4y = 6\) and \(-x + 2y = 3\), doubling the second and adding gives \(0 = 12\): no solution. If you get \(0 = 0\), the equations are multiples of each other and there are infinitely many solutions.
Three variables
Systems with three variables, such as x, y, and z, need three equations. Use elimination to remove one variable from two different pairs, leaving a two-variable system; solve that, then back-substitute. Geometrically, each equation is a plane in space, and a unique solution is the single point where all three meet.
Matrices in brief
Large systems are written as matrices and solved by row reduction, a systematic form of elimination, which is how computers solve systems with thousands of variables in engineering, economics, and graphics. Linear Algebra develops this in depth.
Worked example
A chemist mixes a 10% acid solution with a 30% solution to make 40 litres of 25% solution. With x litres of 10% and y of 30%: \(x + y = 40\) and \(0.10x + 0.30y = 10\). Multiply the second by 10: \(x + 3y = 100\). Subtract the first: \(2y = 60\), so 30 litres of 30% and 10 litres of 10%. Check: \(1 + 9 = 10\) litres of acid.
Key takeaways
Elimination adds or subtracts equations so one variable cancels. Multiply one or both equations first to create opposite or equal coefficients. Subtract when coefficients are equal, add when they are opposite. Use substitution when a variable is easy to isolate and elimination otherwise. If everything cancels, a false statement means no solution and 0 = 0 means infinitely many. Three-variable systems reduce to two-variable ones by repeated elimination, and matrices organise elimination for large systems. Mixture problems lead naturally to two-equation systems.
Flashcard
Add or subtract equations to cancel a variable
Check your understanding
Solve \(x + y = 10\) and \(x - y = 2\).