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.
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., \( \frac{1}{2} \), \( \frac{-3}{4} \)).
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):
- \( x + 5 - 5 = 12 - 5 \)
- \( x = 7 \)
U2 L2 Equation with One Variable - Solve with One Step
Example:
Solve \( x + 3 = 10 \).
Steps:
- Subtract 3 from both sides: \[ x + 3 - 3 = 10 - 3 \]
- Simplify: \[ x = 7 \]
U2 L3 Equation with One Variable - Solve with Two Steps
Example:
Solve \( 2x + 4 = 12 \).
Steps:
- Subtract 4 from both sides: \[ 2x + 4 - 4 = 12 - 4 \] \[ 2x = 8 \]
- Divide both sides by 2: \[ \frac{2x}{2} = \frac{8}{2} \] \[ x = 4 \]
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 \( 3(x + 4) \):
- Multiply: \[ 3 \cdot x + 3 \cdot 4 = 3x + 12 \]
U2 L5 Equation with One Variable on Both Sides
Example:
Solve \( 5x + 2 = 3x + 10 \).
Steps:
- Subtract \( 3x \) from both sides: \[ 5x - 3x + 2 = 10 \] \[ 2x + 2 = 10 \]
- Subtract 2 from both sides: \[ 2x = 8 \]
- Divide by 2: \[ x = 4 \]
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:
- Divide both sides by 10: \[ x = 5 \] Conclusion: You bought 5 tickets.
Feel free to ask more about any of these topics or for further examples!
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