Key Terms

Sure, let's break down the key terms and concepts from your curriculum on expressions and equations. Each section will include a definition, an example with solved problems, and a clear step-by-step explanation.

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### U2 L1 Expression
**Definition:** An expression is a combination of numbers, variables, and operators (such as +, −, ×, ÷) that does not have an equal sign.

**Example:**
Expression: \( 3x + 5 \)

### U2 L1 Terms
**Definition:** A term is a single mathematical expression that can be a number, a variable, or a combination of numbers and variables multiplied together.

**Example:**
Terms in the expression \( 4y^2 + 3y - 5 \):
- \( 4y^2 \) (a term)
- \( 3y \) (a term)
- \(-5\) (a constant term)

### U2 L1 Factors
**Definition:** Factors are numbers or expressions that are multiplied together to yield a product.

**Example:**
In the expression \( 6xy \), the factors are 6, x, and y.

### U2 L1 Coefficients
**Definition:** A coefficient is a numerical factor in a term.

**Example:**
In the term \( 7x^2 \), the coefficient is 7.

**Rational Coefficient?**
- A rational coefficient refers to a coefficient that can be expressed as a fraction (e.g., \( \frac{1}{2} \), \( \frac{-3}{4} \)).

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### U2 L2 Inverse Operations
**Definition:** Inverse operations are operations that undo each other; for example, addition and subtraction are inverse operations, as are multiplication and division.

**Example:**
To solve \( x + 5 = 12 \), you would use the inverse operation of addition (subtraction):
1. \( x + 5 - 5 = 12 - 5 \)
2. \( x = 7 \)

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### U2 L2 Equation with One Variable - Solve with One Step
**Example:**
Solve \( x + 3 = 10 \).

**Steps:**
1. Subtract 3 from both sides:
\[
x + 3 - 3 = 10 - 3
\]
2. Simplify:
\[
x = 7
\]

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### U2 L3 Equation with One Variable - Solve with Two Steps
**Example:**
Solve \( 2x + 4 = 12 \).

**Steps:**
1. Subtract 4 from both sides:
\[
2x + 4 - 4 = 12 - 4
\]
\[
2x = 8
\]
2. Divide both sides by 2:
\[
\frac{2x}{2} = \frac{8}{2}
\]
\[
x = 4
\]

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### U2 L4 Like Terms
**Definition:** Like terms are terms that contain the same variables raised to the same powers.

**Example:**
In the expression \( 3x + 4x - 5 + 2 \):
- Like terms: \( 3x \) and \( 4x \)
- Combine: \( 7x - 5 + 2 = 7x - 3 \)

### U2 L4 Distributive Property
**Definition:** The distributive property states that \( a(b + c) = ab + ac \).

**Example:**
Use the distributive property on \( 3(x + 4) \):
1. Multiply:
\[
3 \cdot x + 3 \cdot 4 = 3x + 12
\]

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### U2 L5 Equation with One Variable on Both Sides
**Example:**
Solve \( 5x + 2 = 3x + 10 \).

**Steps:**
1. Subtract \( 3x \) from both sides:
\[
5x - 3x + 2 = 10
\]
\[
2x + 2 = 10
\]
2. Subtract 2 from both sides:
\[
2x = 8
\]
3. Divide by 2:
\[
x = 4
\]

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### U2 L6 Real World Scenario Setting Up and Solving Equations
**Example:**
A movie costs $10 per ticket. If you buy \( x \) tickets, and you spend a total of $50, how do you find \( x \)?

**Set up the equation:**
\[
10x = 50
\]

**Solve:**
1. Divide both sides by 10:
\[
x = 5
\]
**Conclusion:** You bought 5 tickets.

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Feel free to ask more about any of these topics or for further examples!