The substitution method, explained step by step with a fully worked example, a second harder example, and the mistakes that cost students points on every test.
A system of equations is two equations with the same two variables. The solution is the one point \((x, y)\) that makes both equations true at the same time.
Substitution solves a system by replacing one variable with an equivalent expression from the other equation. The idea is to turn a two-variable problem into a one-variable problem — something much easier to solve.
Isolate one variable in one of the equations. Look for a variable that already has a coefficient of 1 or −1 — it will be the easiest to isolate. Solve for that variable.
Substitute that expression into the other equation. Replace the isolated variable in the second equation with the expression you just found. Now you have one equation with one variable.
Solve the single-variable equation. Use inverse operations to find the value of the remaining variable.
Back-substitute to find the other variable. Plug the value you just found back into the expression from Step 1 to find the second variable. Then check your solution in both original equations.
This one requires isolating the variable yourself first, and has messier numbers.
Both methods solve the same type of problem. Choosing the right one for a given system saves time.
If the substitution leads to a false statement with no variables (like \(3 = 7\)), the system has no solution. The lines are parallel and never intersect.
If substitution leads to a true statement with no variables (like \(5 = 5\)), the system has infinitely many solutions. The equations describe the same line.
Forgetting to distribute. In Step 2, when you substitute an expression like \((3x - 7)\) into a term like \(4(3x - 7)\), you must distribute the 4 to both terms inside the parentheses. Skipping this is the single most common error.
Substituting back into the wrong equation. In Step 4, substitute back into the isolated expression from Step 1, not the original equation you just solved. The isolated form is simpler and less error-prone.
Only finding one variable. The solution is always an ordered pair \((x, y)\). Students sometimes solve for \(x\) and stop — but you need both values, and you need to check both in the original equations.
Sign errors when isolating. When you move a term to isolate a variable, watch the signs carefully. \(3x - y = 7\) → \(-y = 7 - 3x\) → \(y = 3x - 7\). A sign flip here cascades into a wrong answer.
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