Let's break down the key terms one by one, providing definitions, meanings, pictures, and examples with problems.
### 1. Expression
**Definition:**
An expression is a combination of numbers, variables, and mathematical operations (like addition, subtraction, multiplication, and division) that represents a value.
**Meaning:**
An expression does not have an equals sign. It can be simplified or evaluated, but it cannot be solved as an equation can.
**Picture:**
An example of an expression is \(3x + 5\).
**Example Problem:**
Evaluate the expression \(3x + 5\) for \(x = 2\).
**Steps:**
1. Substitute \(x\) with \(2\):
\(3(2) + 5\)
2. Perform the multiplication:
\(6 + 5\)
3. Add:
\(11\)
Final answer: \(11\).
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### 2. Terms
**Definition:**
Terms are the individual components of an expression that are separated by addition or subtraction operators.
**Meaning:**
Each term can be a constant (number), a variable, or a combination of both multiplied together.
**Picture:**
In the expression \(4x^2 + 3x - 7\), the terms are \(4x^2\), \(3x\), and \(-7\).
**Example Problem:**
Identify the terms in the expression \(2x + 5y - 3z + 8\).
**Steps:**
- The terms are \(2x\), \(5y\), \(-3z\), and \(8\).
Final answer: Terms are \(2x\), \(5y\), \(-3z\), \(8\).
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### 3. Factors
**Definition:**
Factors are the quantities we multiply together to get another number or expression.
**Meaning:**
In algebra, factors can be numbers, variables, or both.
**Picture:**
In the expression \(6x\), \(6\) and \(x\) are factors.
**Example Problem:**
Factor the expression \(10xy + 15x\).
**Steps:**
1. Identify the common factor of the terms: The GCF (Greatest Common Factor) is \(5x\).
2. Factor \(5x\) out of each term:
\(5x(2y + 3)\)
Final answer: \(5x(2y + 3)\).
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### 4. Coefficients
**Definition:**
Coefficients are the numerical factors that multiply a variable in a term.
**Meaning:**
They indicate how many times to multiply the variable. If a term does not have an explicit coefficient, it is assumed to be \(1\).
**Picture:**
In the term \(7x\), \(7\) is the coefficient.
**Example Problem:**
Identify the coefficients in the expression \(4x^2 - 3x + 7\).
**Steps:**
- For the first term \(4x^2\): Coefficient is \(4\).
- For the second term \(-3x\): Coefficient is \(-3\).
- For the constant term \(+7\): The coefficient of a constant is not explicitly stated, but we refer to it as \(7\) (it does not have a variable).
Final answer: Coefficients are \(4\), \(-3\), and \(7\).
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### Summary Table
| Term | Definition |
|---------------|------------------------------------------------------------------------|
| Expression | A combination of numbers, variables, and operations representing a value. |
| Terms | Individual parts of an expression separated by '+' or '-' operators. |
| Factors | Numbers or variables that multiply together to produce another number. |
| Coefficients | Numerical factors that multiply the variables in terms. |
These definitions and examples should provide a solid understanding of the key terms related to expressions in algebra! If you have any additional questions, feel free to ask!