Algebra I: 01.02 Solving One-Variable Equations

Questions and Answers

What is an equation?

A mathematical statement containing an equal sign (correct)

A statement that has two different expressions

A number on both sides of the equal sign

An expression that does not contain a variable

What is a solution in the context of an equation?

The value for the variable that makes the equation true

How do you check your answer using substitution?

Substitute the value found for the variable back into the equation.

What is the first step for solving equations containing variables on both sides?

Get the variable on one side by reversing the addition or subtraction of one of the terms. Signup and view all the answers

What are the steps to solve the equation 5x − 2 = 3x + 4?

<ol> <li>Subtract 3x, 2) 2x - 2 = 4, 3) Add 2, 4) 2x = 6, 5) x = 3</li> </ol> Signup and view all the answers

What are the steps to combine like terms?

<ol> <li>Use the distributive property, 2) Combine like terms, 3) Get the variable term alone, 4) Solve for the variable, 5) Check your work.</li> </ol> Signup and view all the answers

What does the Least Common Denominator (LCD) refer to?

The smallest multiple shared by each denominator. Signup and view all the answers

What does 'Given' mean in a mathematical context?

Information provided to solve a problem. Signup and view all the answers

What is the multiplication property of equality?

When both sides of an equation are multiplied by the same number, they remain equal. Signup and view all the answers

What does the Distributive Property state?

A(b + c) = Ab + Ac Signup and view all the answers

What does simplify mean in math?

To write an expression in a simpler form. Signup and view all the answers

What is the addition property of equality?

Adding the same number to both sides of an equation keeps them equal. Signup and view all the answers

What does the subtraction property of equality state?

Subtracting the same number from both sides keeps the equation equal. Signup and view all the answers

What does the division property of equality determine?

Dividing both sides of an equation by a nonzero number keeps them equal. Signup and view all the answers

What is the commutative property of addition or multiplication?

Changing the order of addends or factors does not change the sum or product. Signup and view all the answers

What is the associative property of addition or multiplication?

The way numbers are paired does not affect the sum or product. Signup and view all the answers

What is the reflexive property of equality?

An expression is always equal to itself. Signup and view all the answers

What does the symmetric property of equality allow?

Expressions can be swapped on either side of the equals sign while preserving equality. Signup and view all the answers

What does the transitive property of equality imply?

If a = b and b = c, then a = c. Signup and view all the answers

Study Notes

Key Concepts of Solving One-Variable Equations

Equation: A statement that asserts two expressions are equal, marked by an equal sign.
Solution: The specific value of a variable that satisfies the equation, making it true.
Substitution Check: To verify a solution, replace the variable in the original equation with the found solution and simplify. A true statement confirms correctness.

Solving Equations with Variables on Both Sides

Simplification Step: Rearranging the equation involves isolating the variable by reversing addition or subtraction.
Example Problem: For the equation 5x - 2 = 3x + 4, the process leads to x = 3 after simplification and solving.

Combining Like Terms

Procedure: Involves using the distributive property where necessary, grouping similar terms, isolating the variable, and checking the solution.
Example Problem: The equation 2x + 4 = 4(x - 5) simplifies to x = 12 after combining and solving.

Key Properties in Algebra

Least Common Denominator (LCD): Found by identifying common multiples of given denominators.
Multiplication Property of Equality: States multiplying both sides of an equation by the same number keeps the equality intact.
Distributive Property: Describes how multiplication distributes over addition (a(b + c) = ab + ac).

Definitions of Essential Properties

Addition Property of Equality: Adding the same number to both sides maintains equivalence.
Subtraction Property of Equality: Subtracting the same number from both sides maintains equivalence.
Division Property of Equality: Dividing both sides by the same nonzero number keeps the equation balanced.
Commutative Property: Order of addition or multiplication does not impact the result.
Associative Property: Pairing of numbers during addition or multiplication does not affect the sum or product.

Reflexive, Symmetric, and Transitive Properties

Reflexive Property: Any quantity is equal to itself (e.g., x = x).
Symmetric Property: If one expression equals another, the order can be swapped (e.g., if a = b, then b = a).
Transitive Property: If a = b and b = c, then a must equal c, establishing a linked equality.

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Test your knowledge on one-variable equations with this flashcard quiz. Each card presents key terms and definitions relevant to solving equations. Strengthen your understanding and recall of important concepts in Algebra I.