Algebra 1 — Variables and Expressions (Lesson 1) Flashcards

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Variable

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A symbol (often a lowercase letter) that acts as a placeholder for an unknown quantity. Examples include $x$, $y$, and $z$, and each can take different numerical values in different situations. Variables let you write general expressions and equations that represent many possible numbers.

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Modeling with Variables

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Using variables to represent quantities in real situations so you can write expressions or equations. For example, Mark’s daily pay $63 + x$ uses $x$ for unknown tips; substituting values for $x$ gives specific totals. Modeling makes it easy to compute outcomes for different scenarios.

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Term

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A single number, a variable, or a product of numbers and variables such as $4x$, $9y$, $24xyz$, or $7$. Terms are the building blocks of expressions and are separated by plus or minus signs. Each term has a coefficient (if it contains a variable) and possibly a variable part.

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Coefficient

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The number that multiplies the variable(s) in a term. For example, in $4x^2z$ the coefficient is $4$, and in $x$ the (implicit) coefficient is $1$. Coefficients tell how many copies of the variable part are present.

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Constant

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A number on its own that does not change value, such as the $7$ in $5x + 7$. Constants are terms with no variable part and affect the value of an expression but not its variable behavior. They remain fixed when variables change.

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Algebraic Expression

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One or more terms separated by plus or minus signs, for example $5x + 2y - 4z$. The value of an algebraic expression depends on the values of its variables. Expressions can be simplified, evaluated, and manipulated using algebraic rules.

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Substitution

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The process of replacing each variable in an expression with a given number and then computing the result. For example, for $2x + 3$ with $x=4$ substitute to get $2(4) + 3 = 11$. Always follow the order of operations after substituting values.

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Distributive Property

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A rule that lets you multiply a factor across terms inside parentheses: $a(b + c) = ab + ac$. For instance, $3(x - 4)$ becomes $3x - 12$. Distribution turns products with sums into sums of products and is essential for simplifying and expanding expressions.

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Simplifying Expressions

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The process of using properties like distribution and combining like terms to write an expression in a simpler form. For example, $4(x - 7) + 2$ distributes to $4x - 28 + 2$, which simplifies to $4x - 26$ after combining constants. Simplifying makes expressions easier to evaluate and compare.

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Like Terms

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Terms that have the exact same variable part, including the same letters and exponents, for example $5x$ and $3x$. Like terms can be combined by adding or subtracting their coefficients, but terms like $2x$ and $3x^3$ are not like terms and cannot be combined. Recognizing like terms is key to simplification.

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Combining Like Terms

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Adding or subtracting the coefficients of like terms to produce a single term. For example, $5x + 3x = 8x$ and $4y - 2y = 2y$. You cannot combine unlike terms because their variable parts differ.

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Order of Operations

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The sequence to evaluate expressions: first parentheses and exponents, then multiplication and division (left to right), and finally addition and subtraction (left to right). Following this order ensures consistent results when evaluating expressions. Commonly remembered by mnemonics like PEMDAS.

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Commutative Property

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A property stating that order doesn’t matter for addition and multiplication: $a + b = b + a$ and $ab = ba$. This allows you to rearrange terms or factors without changing the result. The commutative property does not apply to subtraction or division.

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Associative Property

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A property stating that grouping doesn’t affect the result for addition and multiplication: $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$. It lets you regroup terms or factors to simplify calculations. Associativity does not hold for subtraction or division in general.

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Factoring

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The reverse of distribution: identifying a common factor and writing an expression as a product, for example turning $ab + ac$ into $a(b + c)$. Factoring makes solving equations and simplifying expressions easier and will be explored further in later lessons. It helps reveal structure hidden in expanded forms.

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Multiplication Notation

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Writing a number next to a variable means multiplication, for example $2y = 2\times y$. This shorthand (juxtaposition) is common in algebra and should be interpreted as a product. Be careful when substituting so you don’t omit implied multiplication symbols.

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Identifying Terms

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Locate plus and minus signs to separate an expression into its individual terms. Recognizing term boundaries helps you identify coefficients, constants, and like terms for simplification or evaluation. Clear identification prevents mistakes when distributing or combining terms.

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Evaluation Tips

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Always substitute values carefully and then follow the order of operations when computing. Use properties like commutative and associative to rearrange or group terms for easier arithmetic. Double-check sign changes when distributing negatives or combining terms to avoid common errors.