Solving systems of equations
Two equations and two unknowns is one of the most common setups in the section, and the method is never the hard part. The hard part is choosing the method in the first ten seconds instead of grinding through the wrong one for two minutes.
The shape of the problem
A system gives you two facts about the same two quantities. Whatever method you use, the arithmetic ends in one equation with one variable. Read the system for a moment before writing: the shape of the two equations tells you which method will be shorter, and the question tells you whether you need both variables or just one.
Substitution when a variable is alone
If one equation already says y = something, substitution is nearly free. Replace y in the other equation and solve for x, then plug back. The same applies when a variable is almost alone, such as 2x = y + 3, which costs one division to isolate. In a system like y = 2x − 3 and 5x + y = 18, the first line hands you the substitution.
Elimination when the coefficients line up
If the same variable has opposite coefficients in the two equations, add them and it disappears. If the coefficients match, subtract. When they do not match, multiply one equation by a number so that they do. In 3x + 4y = 22 and 3x − 2y = 4, subtracting removes x in one step, which is faster than any substitution.
The combination shortcut
Plenty of questions do not ask for x or y, they ask for x + y or x − y. In those, add or subtract the two equations as they are and see whether the target appears. If 2x + 3y = 17 and x + 2y = 10, then adding gives 3x + 5y = 27, and subtracting gives x + y = 7, which is usually the answer. Solving for both variables first would waste the whole question.
When the system is a word problem
Two unknowns need two sentences. Name the variables with their units before writing anything, then turn each fact into an equation. A group bought 12 tickets for 72 dollars becomes a + c = 12 and 8a + 5c = 72. The count equation is almost always the easy one, and writing it first keeps the money equation honest.
Check the pair, not just the answer
Substitute the pair into both original equations, not only the one you used to solve. A pair that satisfies one equation and fails the other is the most common arithmetic slip in this topic, and the check takes ten seconds.
Which method should I use for a system?
Substitution when a variable is isolated or nearly isolated, elimination when a coefficient matches or is one multiplication away. If the question asks for x + y, add or subtract the equations instead of solving.
How do I decide what to multiply by?
Look for the variable you want to remove, then multiply so its coefficients become opposites. To cancel x in 2x + y = 9 and 3x + 2y = 16, multiply the first equation by 3 and the second by 2.
What if the question asks for x + y?
Add or subtract the two equations and watch for the combination to appear. Solving for both variables works but usually costs a minute you do not need to spend.
How do I know a system has no solution?
The slopes match and the intercepts do not, so the lines are parallel. There is a full guide on that case, linked below.
Related guides
- No solution vs infinite solutions
- Slope, intercepts and parallel lines
- Percent change with multipliers
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