2.1 The Concept of Equality and the Addition Property of Equality
Learning Objectives
- Understand the concept of equality as a balance.
- Verify solutions to equations by substitution.
- Use the addition and subtraction properties of equality to solve equations.
- Simplify both sides of an equation before solving (including using the distributive property).
- Translate word statements into equations and solve.
Introduction
In Chapter 1, we worked with expressions. In this chapter, we begin working with equations.
Recall from an earlier day, you might have an arithmetic problem like:
□ + 4 = 10
This one is easy enough; you would not need to use any official “algebra” to solve it. However, what if we had:
5□ + 4 – 2(□ + 3) = 10
This is too much to think through systematically. We will build equation-solving skills over the following lessons—one skill at a time.
Launch – The Balance Scale Model
Imagine a balance scale.
If both sides weigh the same, the scale is balanced.
If you add or remove weight from one side only, the balance is lost. For example, if we remove 2 lbs from just one side, the scale becomes unbalanced.
Key Idea
Mathematical equations work the same way:
Whatever you do to one side of an equation, you must do to the other side to keep it balanced.
📌 The equal sign (=) means “has the same value as.”
For example: 3 + 2 = 5 and 7 = 4 + 3
Both are true because both sides represent the same value.
The Concept of Equality
An equation is a mathematical statement that two expressions are equal. The equals sign means both sides have the same value.
- Example: x + 5 = 12
- The left side (x + 5) and the right side (12) represent equal quantities.
To keep an equation balanced, we can add or subtract the same value on both sides.
Addition and Subtraction Properties of Equality
If a = b, then a + c = b + c and a – c = b – c.
Adding or subtracting the same value to both sides of an equation keeps it true.
Verifying Solutions to Equations
To verify a solution, substitute the given value into the original equation to check if it makes both sides equal.
Example 2.1.1
Is x = 7 a solution to x + 5 = 12?
Solution
Left side: x + 5 = 7 + 5 = 12
Right side: 12
✅ Both sides equal → x = 7 is a solution.
Example 2.1.2
Is y = 3 a solution to y – 8 = 5?
Solution
Left side: y – 8 = 3 – 8 = –5
Right side: 5
❌ Not equal → y = 3 is not a solution.
Try It
Solving Using the Addition Property of Equality
Example 2.1.3
Solve for x: x + 5 = 12
Solution
Subtract 5 from both sides:
x + 5 – 5 = 12 – 5
x = 7
✅ Check: Substitute x = 7 → 7 + 5 = 12 ✓
Example 2.1.4
Solve for y: –3 + y = 9
Solution
Add 3 to both sides:
–3 + y + 3 = 9 + 3
y = 12
✅ Check: –3 + 12 = 9 ✓
Example 2.1.5
Solve for m: m – (–4) = 9
Solution
Subtracting a negative is the same as adding:
m + 4 = 9
Subtract 4 from both sides:
m = 5
✅ Check: 5 – (–4) = 5 + 4 = 9 ✓
Try It
Solving Equations that Require Simplification
Sometimes we must simplify both sides before using the addition/subtraction properties.
Example 2.1.6
Solve: 9x – 5 – 8x – 6 = 7
Solution
Step 1: Combine like terms on the left side.
(9x – 8x) + (–5 – 6) = 7
x – 11 = 7
Step 2: Add 11 to both sides.
x – 11 + 11 = 7 + 11
x = 18
✅ Check: Substitute x = 18
Left: 9(18) – 5 – 8(18) – 6 = 162 – 5 – 144 – 6 = 7 ✓
Final Answer: x = 18
Example 2.1.7
Simplify and solve: 5(x – 2) = 4x + 1
Solution
Step 1: Apply the distributive property.
5x – 10 = 4x + 1
Step 2: Subtract 4x from both sides.
x – 10 = 1
Step 3: Add 10 to both sides.
x = 11
✅ Check: 5(11 – 2) = 4(11) + 1 → 45 = 45 ✓
Final Answer: x = 11
Example 2.1.8
Simplify and solve: 5 – (x + 6) = –2x + 9
Solution
Step 1: Distribute the negative sign.
5 – x – 6 = –2x + 9
Simplify: –x – 1 = –2x + 9
Step 2: Add 2x to both sides.
x – 1 = 9
Step 3: Add 1 to both sides.
x = 10
✅ Check: 5 – (10 + 6) = –2(10) + 9 → –11 = –11 ✓
Final Answer: x = 10
Try It
Translating Words into Equations
Some equations begin as verbal statements. Recognizing key words helps translate them into algebra.
| Phrase | Translation |
|---|---|
| The sum of a number and 14 | x + 14 |
| A number decreased by 3 | x – 3 |
| Three less than a number | x – 3 |
| The difference of a number and 8 | x – 8 |
| A number increased by 5 | x + 5 |
Example 2.1.9
Translate and solve: “The sum of x and 14 is 22.”
Solution
Translation: x + 14 = 22
Solution: Subtract 14 from both sides.
x = 8
✅ Check: 8 + 14 = 22 ✓
Final Answer: x = 8
Example 2.1.10
Translate and solve: “Three less than x is –10.”
Solution
Translation: x – 3 = –10
Solution: Add 3 to both sides.
x = –7
✅ Check: –7 – 3 = –10 ✓
Final Answer: x = –7
Try It
Common Mistakes
🚩 Doing an operation on only one side — breaks the balance.
🚩 Sign errors — forgetting that subtracting a negative is the same as adding.
🚩 Combining unlike terms — only terms with the same variable and exponent can be combined.
🚩 Forgetting to distribute a negative sign across parentheses.
🚩 Skipping the check step — always verify your answer.
Linear Equations
In this unit, we are only going to focus on linear equations. Linear equations can be written in the form:
Ax + B = C, where A, B, and C are real numbers, and A ≠ 0.
Examples of linear equations: 5x + 2 = 12, 6x – 3x = 8, 10x = 22
Examples of non-linear equations: 4x2 – 6x + 2 = 0, 
, 
Key Takeaways
- An equation is like a balance scale — both sides must stay equal.
- Addition/Subtraction Properties of Equality: add or subtract the same number from both sides to keep balance.
- Always simplify first — combine like terms or distribute before solving.
- Translate word phrases into equations carefully.
- Always check your solution by substitution.
