Lesson 7.3

Substitution (Level 1)

Graphing is messy. Measuring decimals with a ruler is hard. Substitution allows us to find exact answers using pure algebra.

Introduction

In sports, a "Substitution" is when you take one player out and put another player in. In math, if we know , we can take out of any equation and put in its place.

Past Knowledge

Lesson 2.8 (Distribution). You will need to distribute numbers into parentheses frequently.

Today's Goal

Solve systems where one variable is already isolated.

Future Success

This is the best method when you see equations like or .

Key Concepts

The "Blob" Method

Imagine you have a system:

1)

2)

The first equation says that is exactly the same as the "blob" .

So, in the second equation, we can replace with the blob.

The Swap

Now the equation ONLY has s! The is gone. We can solve it.

Worked Examples

Example 1: Simple Number

Basic

Solve the system.

Step 1: Substitute

We know is 3.

Step 2: Solve

Solution:

Example 2: The Blob

Intermediate

Solve the system.

Step 1: Substitute

Replace with in the second equation.

Step 2: Distribute & Solve

Step 3: Plug Back In (The Boomerang)

We aren't done! We need .

Use the first equation:

Solution:

Example 3: Isolated X

Advanced

Solve the system.

Step 1: Substitute for X

This time is isolated. Plug it into the spot.

Step 2: Solve

Step 3: Boomerang

Solution:

Common Pitfalls

Forgetting Parentheses

This is the #1 mistake. When you substitute , you MUST wrap it in parentheses . If you don't, you won't distribute correctly.

Solving for X, Stopping

Finding the first variable is only half the battle. You need the full coordinate pair . Always plug your answer back in to find the buddy variable.

Real-Life Applications

Computer Programming:

  • In coding, we use variables constantly.
  • let taxRate = 0.05;
  • let total = price + (price * taxRate);
  • The computer substitutes the value 0.05 wherever it sees "taxRate". You are learning the logic of how software thinks!

Practice Quiz

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