Expression vs. Equation

Algebra-1

1. Fundamental Concepts

Definition: An expression is a combination of numbers, variables, and operations without an equals sign. For example, $$3x + 5$$ .
Definition: An equation is a statement that two expressions are equal. It contains an equals sign. For example, $$3x + 5 = 14$$ .
Key Difference: Expressions represent values, while equations assert relationships between values.

2. Key Concepts

Evaluating Expressions: Substitute values for variables and simplify the expression.

Solving Equations: Find the value(s) of the variable(s) that make the equation true.

Application: Expressions and equations are used in various real-world scenarios, such as calculating costs, distances, and more.

3. Examples

Example 1 (Basic)

Problem: Evaluate the expression $$2x + 3$$ when $$x = 4$$ .

Step-by-Step Solution:

Substitute $$x = 4$$ into the expression: $$2(4) + 3$$
Calculate: $$8 + 3 = 11$$

Validation: Substitute $$x = 4$$ → Original: $$2(4) + 3 = 11$$ ; Simplified: $$8 + 3 = 11$$ ✓

Example 2 (Intermediate)

Problem: Solve the equation $$3x - 7 = 11$$ .

Step-by-Step Solution:

Add 7 to both sides: $$3x - 7 + 7 = 11 + 7$$
Simplify: $$3x = 18$$
Divide both sides by 3: $$\frac{3x}{3} = \frac{18}{3}$$
Solve for x: $$x = 6$$

Validation: Substitute $$x = 6$$ → Original: $$3(6) - 7 = 11$$ ; Simplified: $$18 - 7 = 11$$ ✓

4. Problem-Solving Techniques

Isolate Variable: Move all terms with the variable to one side of the equation and all other terms to the other side.
Check Solutions: Always substitute your solution back into the original equation to verify it.
Use Parentheses: Use parentheses to group terms and avoid mistakes when substituting values.