Substitution Method Calculator
This substitution method calculator solves a system of linear equations the way the method is actually taught — and shows the route rather than just the destination.
Take 2x + 3y = 12 with x − y = 1. The second equation hands over x = y + 1 for free, since x already has a coefficient of 1 and nothing has to be divided. Put that in place of x in the first equation and it reads 2(y + 1) + 3y = 12, which flattens to 5y = 10, so y = 2. Back into x = y + 1 gives x = 3. Four lines, and every one of them appears above.
How the Substitution Method Works
Four moves, always in the same order. Choose a letter and make it the subject of one equation. Carry that expression into the other equation, in brackets. Solve what is left, which now holds a single unknown. Put that value back into the expression from the first move.
If one of your equations already reads y = 3x − 2 or anything of that shape, the first move is done for you and the question is half as long. That is exactly when a marker expects substitution rather than any other route.
The reason it is allowed is worth a sentence, because it is the whole idea. An equation is a statement that two things are the same, so wherever one appears the other may stand in its place. Writing x = y + 1 earns the right to write y + 1 anywhere x was.
Which Letter to Isolate
This is the decision that separates a tidy page of working from a messy one, and it is rarely taught explicitly. Look for a coefficient of 1: that letter comes out with no division at all. A coefficient of −1 is next best, since only signs change. Failing both, take the smallest whole number, because that is the divisor you will be carrying for the rest of the question.
When nothing is easy the fractions start immediately. Solving 3x + 2y = 7 for y gives y = −(3/2)x + 7/2, and substituting into 5x − 4y = 3 gives 11x = 17, so x = 17/11 and y = 13/11. Those are the exact answers. Nothing is rounded anywhere in this calculator, which matters here: a decimal that has been trimmed twice can turn an exact fraction into an answer that fails the check.
When the Letters Cancel
Sooner or later a substitution wipes out every letter and leaves nothing but numbers. Students usually assume they have gone wrong. They have not — that line is the answer.
Try x + y = 2 with x + y = 5. The first gives x = 2 − y, and putting that into the second reads (2 − y) + y = 5, which collapses to 0 = 3. That can never be true, so no pair of values satisfies both equations: the system has no solution.
Now try x + y = 2 with 2x + 2y = 4. The same first step gives 2(2 − y) + 2y = 4, which collapses to 0 = 0. That is true whatever the letters are, which tells you the second equation repeated the first and added nothing. Infinitely many pairs work. Most calculators print the verdict and hide the line; this one prints the line, because the line is where the reasoning lives.
Substitution With Three Variables
The method does not change, it simply runs twice. Isolate one letter, substitute it into both remaining equations, and three unknowns become two; isolate again and two become one. Given x + y + z = 6, 2y + 5z = −4 and 2x + 5y − z = 27, the calculator works down to z, then walks back through y to x, arriving at x = 5, y = 3, z = −2. Most substitution tools stop at two unknowns; this one does not.
How to Use This Substitution Method Calculator
Each equation has two boxes with the equals sign printed between them, so you never type one. Nothing needs rearranging first: if your book already gives y = 3x − 2, put y in the left box and 3x − 2 in the right. Fractions, decimals and brackets are all accepted, and the keypad carries a, b and c as well as x, y and z.
Press = and open Show the substitution. You get the numbered route: the letter chosen and the reason, the expression carried in with the bracket picked out, the shorter equation it leaves, and the walk back. Before any answer is shown, the values are put back into the equations you typed and both sides are compared exactly. If you only want the answer and a picture of where the lines meet, the system of equations calculator does that instead.
Substitution Method FAQ
What is the substitution method?
Which variable should I isolate first?
What happens when the variables cancel out?
Can the substitution method be used with three variables?
Is substitution still worth using when no coefficient is 1?
What does back-substitution mean?
Free the cheapest letter, carry it across, solve what is left, walk it back. Everything the substitution method asks of you is in those four moves — and the line where the letters vanish is a result, not a wrong turn.