Proving Why Your Steps Are Valid

Explaining Each Step in Solving an Equation

In this lesson:

• Name the properties of equality and explain why each works
• Write justified solution steps in two-column format
• Understand the logical structure behind equation solving

What You Will Be Able to Do

1. Name and apply the four properties of equality
2. Justify each solving step by citing the property
3. Explain the IF-THEN logical structure of equation solving
4. Write a justified solution in two-column format
5. Construct a viable argument for a solution method

You Know the Steps — Prove Them

You've solved before:

• Subtract 5 → ; divide by 3 →

But WHY is subtracting from both sides a legal move?

Today we build the real answer.

The Balance Scale: A Physical Test

Imagine a balance scale with the same object on both pans.

What keeps the scale balanced — and what breaks it?

Addition and Subtraction Properties of Equality

• Addition: If , then
• Subtraction: If , then

Same value on both sides → equality preserved (equal weights, both pans).

Multiplication and Division Properties of Equality

• Multiplication: If , then
• Division: If , : then

Multiplying by 0: equation becomes — all information lost

Dividing by a variable: confirm nonzero first

Quick Check: Name That Property

For each algebraic step, name the property used:

1. → — which property?
2. → — which property?
3. → — which property?

Write your answers before advancing.

Two-Column Format Makes Reasoning Visible

Two-Column Walkthrough with Properties Named

• Subtract from both sides → Subtraction Property
• Add to both sides → Addition Property
• Divide both sides by → Division Property

Distributive Property Before Equality Properties

Solve :

Step 1: — Distributive + Simplify

Step 2: — Subtraction Property

Step 3: — Subtraction then Division Property

Fill in the Reasons: Your Turn

— supply the reasons:

The Ledger Is Complete — Now What?

The two-column proof ends with . Has this proven is a solution?

• The derivation assumed a solution exists
• It derived what that solution must be
• The check confirms the assumption was valid

This IF-THEN structure is what equation solving actually is.

IF a Solution Exists, THEN It Must Be...

Every equation solution is a chain of logical implications:

Each arrow means: any solution to the equation above must also satisfy the equation below.

Verification Confirms the Logical Circle

The check closes the argument:

• Derivation: IF a solution exists, it must equal
• Check: satisfies the original → confirmed

Without the check: only proved "if a solution exists, it's 3."

When the Derivation Ends Without a Variable

Sometimes a correct derivation gives a surprising result:

• False statement (): No value of satisfies the equation → no solution
• True statement (): Every value of satisfies the equation → infinite solutions

These are not errors — they ARE the proof.

Solving an Equation with No Solution

Solve :

is false → no solution. This IS the proof.

Quick Check: What Does Each Result Mean?

You reach one of these after valid steps:

• Case A:
• Case B:
• Case C:

What does each tell you about the solution set? Write before advancing.

Justifying Method Choice, Not Just Steps

A viable argument defends your approach, not just individual steps:

• Step: "Subtracted — Subtraction Property"
• Method: "Subtracted first to eliminate the variable from the right side, isolating most efficiently"

The standard requires both.

Two Approaches to the Same Equation

Solve two ways:

Approach A — Distribute first: expand , then collect variable terms

Approach B — Divide first: divide both sides by 3, then expand

Both are valid. Which is cleaner? Why?

Clear Decimals First — A Strategic Choice

For :

Approach A: Work with decimals → divide by →

Approach B: Multiply both sides by 4 → →

Write your one-sentence argument for your preferred approach.

Produce a Complete Justified Solution

Solve from scratch:

Fill every row. Verify your answer. Write your method argument.

Identify the Property Error in Row 2

Solving , row 2 contains an error:

— "Subtraction Property" ← ERROR

Rows 3–4 continue: →

Name the violation. Write the correct row 2.

Compare Methods, Defend Your Choice

Solve :

A: Distribute first → →

B: Divide by 2 first → →

Defend your choice in one sentence.

Three Errors to Watch For

Different operations on each side — always the SAME operation on BOTH sides

"Distributive Property of Equality" — not real; Distributive rewrites ONE side

Treating as a mistake — it IS the correct result: no solution exists

Equation Solving Is Logical Argument

✓ Properties of equality are the axioms making each step valid

✓ Two-column format makes every implication visible

✓ IF a solution exists the steps derive what it must be

✓ No solution / infinite solutions — valid proof outcomes

Equality: BOTH sides. Distributive: ONE side.

Next Lesson: When Verification Fails

In HSA.REI.A.2, the same IF-THEN reasoning applies to radical and rational equations.

New challenge: A valid derivation sometimes produces a value that fails the check — an extraneous solution.

Today's IF-THEN logic explains exactly why this happens.