When the Variable Disappears

One Solution, No Solution, or Infinitely Many

In this lesson:

• Solve any linear equation to its terminal form
• Read the form to find how many solutions exist

When the Variable Disappears Entirely

You expect to solve and get a number:

But sometimes the vanishes and you get or .

Is that an error — or is it information?

What You Will Learn To Do Today

By the end of this lesson, you should be able to:

1. Solve a linear equation step by step to a terminal form
2. Identify the three terminal forms: , , and
3. Classify an equation as one, infinitely many, or no solution
4. Build your own equations of each solution type
5. Explain why an identity or a contradiction behaves as it does

Solve One You Already Trust

• Subtract from both sides:
• Add to both sides:
• Divide by :

We reached the form — the terminal form.

Three Possible Destinations for Any Equation

Every linear equation lands on exactly one of these three.

An Equation That Cancels to True

• Distribute the left side:
• Subtract from both sides:

The variable is gone, and is always true.

Why "Always True" Means Every Answer

When you reach , the two sides were the same expression.

• The equation is true no matter what is
• So every value of is a solution
• This is called an identity

Test It — Does Every Value Work?

For , try a few values:

Can you find any value that fails?

An Equation That Cancels to False

• Subtract from both sides:

The variable is gone, and is never true.

"Did I Make a Mistake?" — No

Getting feels like an error. Check the arithmetic — it's correct.

• A false statement means no value of works
• This is called a contradiction
• The left side is always more than the right

The Map From Form to Solutions

Your Turn: Classify These Two

Solve completely, then name the form and the count:

Distribute, combine, simplify. What's left when the dust settles?

Collapse Any Equation to One Form

After moving to one side, every linear equation becomes:

• Is the coefficient zero or not?
• If , is also zero?

Two questions decide everything.

The Three Cases of ax = b

• If : divide by → one solution
• If and : → infinitely many
• If and : is false → none

Build a One-Solution and an Identity

One solution — start from , build outward:

Identity — write one expression two ways:

Build a Contradiction on Purpose

Match the variable terms, mismatch the constants:

• Left distributes to
• Same , but → no solution

Two Lines Tell Three Different Stories

Each side of the equation is a line — how they meet is the answer.

Both Sides Doesn't Mean Special

Predict the solution count for each:

All three have variables on both sides. Do they behave the same?

Quick Check: Predict, Don't Solve

On your own, write your prediction for each:

One word each: one, infinitely many, or none. Commit before the reveal.

Find the Error in This Solution

A student solves and writes:

• , so

Where did the type flip from "none" to "one"? Find the step.

Build All Three Types Yourself Now

Write one equation for each type. Each must have:

• At least one set of parentheses
• A variable on both sides

One solution. Infinitely many. None. Then check each by solving.

Four Common Traps to Avoid Here

does not mean — it means every value works
A false result like is no solution, not a mistake
Variables on both sides ≠ special — matching coefficients is the key
"Infinitely many" means every real number, not just many

What You Now Understand About Solutions

✓ Every linear equation lands on , , or
✓ The terminal form is the count: one, all, or none
✓ A vanished variable is information, not an error

Coming Up Next: Solving Systems

You can now classify a single equation's solutions.

Next you'll graph two lines together and solve systems — where one solution, no solution, and infinitely many show up as crossing, parallel, and identical lines (8.EE.C.8).