In short
The substitution method turns a two-variable problem into a one-variable problem in two clean strokes. Step 1 — isolate: pick one equation, isolate one variable (say y), so it sits alone on the left as an expression in x. Step 2 — inject: wherever y appears in the other equation, replace it by that expression. The other equation now has only x in it. Solve for x, then back-substitute into the isolated form to get y. It is one of the three standard methods for CBSE Class 10 (the other two being graphical and elimination), and it works because replacing equals with equals is always legal.
You are staring at two equations:
Each one alone is an entire line — infinitely many (x,y) pairs satisfy it. But the crossing point, the single (x, y) that satisfies both at once, is what you want. The substitution method finds it by collapsing the system into a single equation in a single variable. Here is the trick, in two moves.
The two moves
Move 1 — isolate. Take one equation and rearrange it so a chosen variable sits alone on one side. From x + y = 7, isolate y: \,y = 7 - x. Why: subtracting x from both sides preserves equality — both lines, before and after, have the same set of solutions.
Move 2 — inject. The expression 7 - x is now a certified replacement for y — anywhere y appears, you can write 7 - x instead, and the truth value will not change. So in the other equation, 2x - y = 5, replace y by 7 - x:
This new equation has only x in it. Why: the moment you injected 7 - x in place of y, every reference to y vanished from the second equation. One variable left, and one equation — exactly enough to solve.
Solve: 2x - 7 + x = 5 \implies 3x = 12 \implies x = 4. Back-substitute into y = 7 - x to get y = 3. The crossing point is (4, 3). Done.
Watch the variable collapse
Click the buttons. Equation 1 isolates y, then the isolated expression slides over and replaces the y in Equation 2.
The flow, drawn
Worked examples
Example 1 — the clean case
Solve the system \{\,x + y = 7,\ 2x - y = 5\,\}.
Isolate y in equation 1: y = 7 - x.
Inject into equation 2:
Back-substitute: y = 7 - 4 = 3.
Solution: (x, y) = (4, 3). Check: 4 + 3 = 7 and 2(4) - 3 = 5. Both satisfied.
Example 2 — already isolated
Solve \{\,2x + 3y = 13,\ x = y + 1\,\}.
The second equation already has x isolated — no work needed in step 1. Inject x = y + 1 into equation 1:
Back-substitute: x = y + 1 = \tfrac{11}{5} + 1 = \tfrac{16}{5}.
Solution: \left(\tfrac{16}{5},\ \tfrac{11}{5}\right). Why this case is fast: when one equation is already in x = \ldots or y = \ldots form, you skip the isolation step and go straight to injection.
Example 3 — when substitution is messy
Solve \{\,3x + 4y = 11,\ 2x - 5y = 8\,\}.
Neither variable isolates cleanly — every isolation produces a fraction. Push through anyway. From equation 1, isolate x:
Inject into equation 2:
Multiply through by 3 to clear the denominator:
Then x = \dfrac{11 - 4(-2/23)}{3} = \dfrac{11 + 8/23}{3} = \dfrac{(253 + 8)/23}{3} = \dfrac{261}{69} = \dfrac{87}{23}.
Solution: \left(\tfrac{87}{23},\ -\tfrac{2}{23}\right). It works — but the arithmetic is ugly. For a system like this, elimination (multiply equation 1 by 5 and equation 2 by 4, then add to kill y) is usually faster. Substitution is at its best when at least one variable is easy to isolate; otherwise consider switching tools.
Why this is allowed at all
The whole method rests on one principle: if A = B, then anywhere you see A, you may write B instead, and any true statement stays true. Why: that is literally what the equals sign means — A and B are the same number. Replacing equals by equals never invents or destroys solutions.
So when you say "y equals 7 - x" and then write 7 - x in place of y, you are not approximating, not guessing — you are using the certificate that the first equation hands you. Every (x, y) that satisfies both originals also satisfies the collapsed equation, and every x that solves the collapsed equation pairs (via back-substitution) with exactly one y that satisfies both originals.
When to reach for substitution
- Use it when one equation already has a variable isolated, or when isolation produces small integers (no fractions).
- Skip it when both equations have ugly coefficients on every variable — elimination usually wins there.
- In CBSE Class 10, all three methods (graphical, substitution, elimination) are tested. Substitution is your fastest pencil tool when one equation looks like y = mx + c or x = \ldots already.
References
- NCERT Class 10 Mathematics, Chapter 3: Pair of Linear Equations in Two Variables — the official Indian syllabus treatment.
- Substitution method (Wikipedia) — formal statement and worked examples.
- Khan Academy: Solving systems by substitution — additional practice problems.