Meaning - Definition - Picture

Question

Meaning - Definition - Picture
key terms:
Expression
Terms
Factors
Coefficients 
Inverse operations
Equation with one variable solve with one step
Equation with one variable solve with two steps
Like terms
Distributive property
please give examples with problems solved showing all of the steps.

GPT-4o mini · Answer

Here’s 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 \(x\).

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 \(-7\).

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: \[ 2x(x + 4) \]

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 \(7\).

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: \[ x + 5 - 5 = 12 - 5 \implies x = 7 \]

6. Equation with One Variable Solved with One Step

Example: Solve \(x + 7 = 10\).

Steps:

Subtract \(7\) from both sides (inverse operation of addition): \[ x + 7 - 7 = 10 - 7 \]
Simplify: \[ x = 3 \]

7. Equation with One Variable Solved with Two Steps

Example: Solve \(2x + 3 = 11\).

Steps:

Subtract \(3\) from both sides: \[ 2x + 3 - 3 = 11 - 3 \] \[ 2x = 8 \]
Divide by \(2\): \[ \frac{2x}{2} = \frac{8}{2} \] \[ x = 4 \]

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:

Combine like terms: \[ 3x + 2x + 5 = 5x + 5 \]

9. Distributive Property

Definition: The distributive property states that \(a(b + c) = ab + ac\).

Example: Simplify \(3(x + 4)\).

Steps:

Distribute \(3\) to both terms inside the parentheses: \[ 3(x) + 3(4) = 3x + 12 \]

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!

Answer

you need to involve the meaning of each key-term

GPT-4o mini · Answer

Certainly! Below is a comprehensive explanation of each key term, including its meaning and a related example that showcases the term in context.

1. Expression

Meaning: An expression is a mathematical phrase that combines numbers, variables, and operations (like addition and multiplication) but does not include an equality sign. It represents a value but is not an equation.

Example: \[ 3x + 5 \]

Steps to Evaluate (for a specific value, say \(x = 2\)):

Replace \(x\) with \(2\): \[ 3(2) + 5 \]
Perform the multiplication: \[ 6 + 5 \]
Add: \[ 11 \]

2. Terms

Meaning: Terms are the individual components of an expression that are separated by addition or subtraction. Each term can be a constant, a variable, or a product of both.

Example: In the expression \(4x^2 + 3x - 7\), the terms are:

\(4x^2\)
\(3x\)
\(-7\)

3. Factors

Meaning: Factors are the quantities that are multiplied together to form a product. Recognizing factors in expressions can help simplify or solve equations.

Example: For the expression \(2x^2 + 8x\), you can factor out the common term: \[ 2x(x + 4) \]

Steps:

Identify the common factor: \(2x\).
Factor it out: \[ 2x(x + 4) \]

4. Coefficients

Meaning: A coefficient is a numerical value that is multiplied by a variable in a term. It indicates how many units of the variable are present.

Example: In the term \(7y\), the coefficient is \(7\) because it indicates that you have \(7\) times the variable \(y\).

5. Inverse Operations

Meaning: Inverse operations are pairs of operations that undo each other. The main pairs are addition and subtraction, and multiplication and division.

Example: To solve the equation \(x + 5 = 12\), you can use subtraction as the inverse of addition.

Steps:

Subtract \(5\) from both sides: \[ x + 5 - 5 = 12 - 5 \]
Simplify: \[ x = 7 \]

6. Equation with One Variable Solved with One Step

Meaning: This refers to an equation that can be resolved in a single arithmetic step by using inverse operations.

Example: Solve \(x + 7 = 10\).

Steps:

Subtract \(7\): \[ x + 7 - 7 = 10 - 7 \]
Thus, \(x = 3\).

7. Equation with One Variable Solved with Two Steps

Meaning: This refers to an equation that requires two arithmetic operations to isolate the variable.

Example: Solve \(2x + 3 = 11\).

Steps:

Subtract \(3\) from both sides: \[ 2x + 3 - 3 = 11 - 3 \] \[ 2x = 8 \]
Divide both sides by \(2\): \[ \frac{2x}{2} = \frac{8}{2} \] \[ x = 4 \]

8. Like Terms

Meaning: Like terms are terms in an expression that share the same variable(s) raised to the same power, allowing them to be combined.

Example: In the expression \(3x + 5 + 2x\), \(3x\) and \(2x\) are like terms.

Steps to Combine:

Combine the like terms: \[ 3x + 2x + 5 = (3 + 2)x + 5 = 5x + 5 \]

9. Distributive Property

Meaning: The distributive property states that multiplying a number by a sum (or difference) is the same as multiplying that number by each addend (or subtrahend) individually and then adding (or subtracting).

Example: Simplify \(3(x + 4)\).

Steps:

Distribute \(3\) to both \(x\) and \(4\): \[ 3 \cdot x + 3 \cdot 4 = 3x + 12 \]

These explanations provide definitions and meanings for each key term while also showing how they can be applied in mathematical problems. If you have further questions or need more clarification, just let me know!