Look at these two definitions in the following sections and compare the examples to ensure you know the distinction between an expression and an equation.

Defining an Algebraic Expression

An **algebraic expression** is a collection of constants, variables, symbols of operations, and grouping symbols, as shown in Example 1.

**Example 1:** 4( *x* − 3) + 6

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**Defining an Algebraic Equation**

**An algebraic equation** is a statement that two algebraic expressions are equal, as shown in Example 2.

**Example 2:** 4( *x* − 3) + 6 = 14 + 2 *x*

The easiest way to distinguish a math problem as an equation is to notice an equals sign.

In Example 3, you take the algebraic expression given in Example 1 and simplify it to review the process of simplification. An algebraic expression is simplified by using the **distributive property** and combining **like terms.**

**Example 3:** Simplify the following expression: 4( *x* − 3) + 6

Here is how you simplify this expression:

**1. Remove the parentheses using the distributive property.**

4 *x* + −12 + 6

**2. Combine like terms.**

The simplified expression is 4 *x* + −6.

**Note**: This problem does not solve for *x*. This is because the original problem is an expression, not an equation, and, therefore, cannot be solved.

In order to solve an equation, follow these steps:

**1. Simplify both sides of the equation by using the distributive property and combining like terms, if possible.**

**2. Move all terms with variables to one side of the equation using the addition property of equations, and then simplify.**

**3. Move the constants to the other side of the equation using the addition property of equations and simplify.**

**4. Divide by the coefficient using the multiplication property of equations.**

In Example 4, you solve the equation given in Example 2, using the four preceding steps to find the solution to the equation.

**Example 4:** Solve the following equation: 4( *x* − 3) + 6 = 14 + 2 *x*

Use the four steps to solving a linear equation, as follows:

- 1.

**Distribute and combine like terms.**

- 2a.

**Move all terms with variables to the left side of the equation.**

In this example, add a **−2x** to each side of the equation.

The addition property of equations states that if the same term is added to both sides of the equation, the equation remains a true statement. The addition property of equations also holds true for subtracting the same term from both sides of the equation.

- 2b.

**Place like terms adjacent to each other and simplify.**

**Note:** Subtracting 6 is changed to adding −6 because the commutative property of addition works only if all operations are addition.

- 3.

**Move the constants to the right side of the equation and simplify.**

**Note:** The opposite operation was used to move the constant.

- 4.

**Divide by the coefficient and simplify.**

The solution is *x* = 10.

**Example 5:** Solve the following equation: 12 + 2(3 *x* − 7) = 5 *x* − 4

Use the four steps to solving a linear equation, as follows:

- 1a.

**Distribute and combine like terms.**

- 1b.

**Place like terms adjacent to each other and simplify.**

- 2a.

**Move variables to the left side of the equation.**

In this example, add −5 *x* to each side of the equation.

- 2b.

**Place like terms adjacent to each other and simplify.**

**Note:** All subtractions are changed to addition of a negative number.

- 3.

Move the constants to the right side of the equation and simplify.

**Note:** The opposite operation was used to move the constant.

- 4.

**Because the coefficient is 1, Step 4 is not necessary.**

The solution is *x* = −2.

**Example 5:** Solve the following equation: 6 − 3(2 − *x*) = −5 *x* + 40

Use the four steps to solving a linear equation, as follows:

- 1.

**Distribute and combine like terms.**

Did you remember to distribute the negative three?

- 2a.

**Move variables to the left side of the equation.**

In this example, add 5 *x* to each side of the equation.

- 2b.

**Place like terms adjacent to each other.**

- 2c.

**Simplify by combining like terms.**

- 3.

**This step is not necessary in this example because all of the constants are on the right side of the equation.**

- 4.

**Divide by the coefficient and simplify.**

The solution is *x* = 5.

**Remember:** The four steps for solving equations must be done in order, but not all steps are necessary in every problem.