Substitution

Algebra

process

Also known as: replacing variables, plug in value, sub in

Grade 6-8

Replacing every occurrence of a variable or sub-expression with an equivalent value or expression throughout a problem. Key technique for solving systems and simplifying expressions.

This concept is covered in depth in our substitution method for systems, with worked examples, practice problems, and common mistakes.

Definition

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

💡 Intuition

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

🎯 Core Idea

Substitution exploits equality—equal things can replace each other.

Example

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

Formula

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

Notation

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

🌟 Why It Matters

Key technique for solving systems and simplifying expressions.

💭 Hint When Stuck

Use parentheses around the entire expression you are substituting in, then simplify step by step.

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)).

Related Concepts

Equations Equal Systems of Equations

🚧 Common Stuck Point

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

⚠️ 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

Go Deeper

Worked Examples

Step-by-step solved problems

Practice Problems

Test your understanding

Formula Explained

Notation, derivation, and common mistakes

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

Frequently Asked Questions

What is Substitution in Math?

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

Why is Substitution important?

Key technique for solving systems and simplifying expressions.

What do students usually 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 Substitution?

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

Prerequisites

equations equal

Next Steps

systems of equations

Cross-Subject Connections

u substitution — Integration by substitution extends algebraic substitution trigonometric functions — Trig substitution simplifies integrals with radicals

How Substitution Connects to Other Ideas

To understand substitution, you should first be comfortable with equations and equal. Once you have a solid grasp of substitution, you can move on to systems of equations.

Want the Full Guide?

This concept is explained step by step in our complete guide:

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

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