Substitution Formula

The Formula

If y = g(x), then f(y) = f(g(x))

When to use: If y = 2x, you can write 2x everywhere you see y—they're the same.

Quick Example

Given y = x + 3, substitute into 2y: 2(x + 3) = 2x + 6

Notation

Substitution is written 'let y = \ldots' or 'substitute y = \ldots into.' Parentheses around the substituted expression are essential.

What This Formula Means

Replacing every occurrence of a variable or sub-expression with an equivalent value or expression throughout a problem.

If y = 2x, you can write 2x everywhere you see y—they're the same.

Formal View

If y = g(x), then for any expression f(y), the substitution y \mapsto g(x) yields f(g(x)). Formally, this is function composition: (f \circ g)(x) = f(g(x)).

Worked Examples

Example 1

easy
If y = 3x + 1 and x = 2, find y.

Solution

  1. 1
    Replace x with 2 in the expression: y = 3(2) + 1.
  2. 2
    Compute: y = 6 + 1 = 7.
  3. 3
    Substitution replaces a variable with its known value.

Answer

y = 7
Substitution means replacing a variable with an equivalent value or expression. Here we replace x with 2 to find the corresponding y.

Example 2

medium
If y = x + 3 and 2x + y = 9, use substitution to solve for x.

Common Mistakes

  • Replacing only one occurrence of the variable instead of every occurrence in the expression
  • Dropping parentheses when substituting a multi-term expression — writing 2 \cdot x + 3 instead of 2(x + 3)
  • Substituting the wrong direction — if y = 2x, putting y in place of 2x when asked to eliminate y

Why This Formula Matters

Key technique for solving systems and simplifying expressions.

Frequently Asked Questions

What is the Substitution formula?

Replacing every occurrence of a variable or sub-expression with an equivalent value or expression throughout a problem.

How do you use the Substitution formula?

If y = 2x, you can write 2x everywhere you see y—they're the same.

What do the symbols mean in the Substitution formula?

Substitution is written 'let y = \ldots' or 'substitute y = \ldots into.' Parentheses around the substituted expression are essential.

Why is the Substitution formula important in Math?

Key technique for solving systems and simplifying expressions.

What do students get wrong about Substitution?

You must substitute the value into EVERY occurrence in the expression, not just the first one you see.

What should I learn before the Substitution formula?

Before studying the Substitution formula, you should understand: equations, equal.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Solving Systems of Equations: Substitution, Elimination, and Matrices →
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