Hereโs a breakdown of your key terms along with their definitions, meanings, illustrations, and examples, showing step-by-step problem-solving.
### 1. Expression
**Definition:** An expression is a combination of numbers, variables (letters that represent numbers), and operations (like addition and multiplication) but does not have an equality sign.
**Example:**
\[ 3x + 5 \]
**Steps:**
This expression can't be solved further without additional information, but it can be simplified by combining like terms or evaluated for specific values of \(x\).
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### 2. Terms
**Definition:** Terms are the individual parts of an expression, which could be numbers (constants), variables, or a combination of both multiplied together.
**Example:**
In the expression \(4x^2 + 3x - 7\), the terms are \(4x^2\), \(3x\), and \(-7\).
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### 3. Factors
**Definition:** Factors are the numbers or expressions that are multiplied together to get another number or expression.
**Example:**
Factors of the expression \(2x^2 + 8x\) can be expressed as:
\[ 2x(x + 4) \]
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### 4. Coefficients
**Definition:** A coefficient is a numerical part of a term that is multiplied by a variable.
**Example:**
In the term \(7y\), the coefficient is \(7\).
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### 5. Inverse Operations
**Definition:** Inverse operations reverse the effect of an operation. The two key pairs are addition and subtraction, and multiplication and division.
**Example:**
If you have the equation \(x + 5 = 12\), to solve for \(x\), you perform the inverse operation of addition, which is subtraction:
\[ x + 5 - 5 = 12 - 5 \implies x = 7 \]
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### 6. Equation with One Variable Solved with One Step
**Example:**
Solve \(x + 7 = 10\).
**Steps:**
1. Subtract \(7\) from both sides (inverse operation of addition):
\[
x + 7 - 7 = 10 - 7
\]
2. Simplify:
\[
x = 3
\]
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### 7. Equation with One Variable Solved with Two Steps
**Example:**
Solve \(2x + 3 = 11\).
**Steps:**
1. Subtract \(3\) from both sides:
\[
2x + 3 - 3 = 11 - 3
\]
\[
2x = 8
\]
2. Divide by \(2\):
\[
\frac{2x}{2} = \frac{8}{2}
\]
\[
x = 4
\]
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### 8. Like Terms
**Definition:** Like terms are terms that have the same variable raised to the same power.
**Example:**
In the expression \(3x + 5 + 2x\), the terms \(3x\) and \(2x\) are like terms.
**Steps to Combine:**
1. Combine like terms:
\[
3x + 2x + 5 = 5x + 5
\]
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### 9. Distributive Property
**Definition:** The distributive property states that \(a(b + c) = ab + ac\).
**Example:**
Simplify \(3(x + 4)\).
**Steps:**
1. Distribute \(3\) to both terms inside the parentheses:
\[
3(x) + 3(4) = 3x + 12
\]
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These definitions and examples illustrate how to approach these mathematical concepts systematically. If you have any further questions or need additional examples, feel free to ask!