2.S: Introduction to the Language of Algebra (Summary)
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Key Terms
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coefficient
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The constant that multiplies the variable(s) in a term.
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composite number
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A composite number is a counting number that is not prime.
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divisibility
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If a number m is a multiple of n, then we say that m is divisible by n.
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equation
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An equation is made up of two expressions connected by an equal sign.
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evaluate
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To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number.
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expression
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An expression is a number, a variable, or a combination of numbers and variables and operation symbols.
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least common multiple (LCM)
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The smallest number that is a multiple of two numbers.
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like terms
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Terms that are either constants or have the same variables with the same exponents.
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multiple of a number
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A number is a multiple of n if it is the product of a counting number and n.
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prime factorization
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The product of prime numbers that equals the number.
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prime number
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A counting number greater than 1 whose only factors are 1 and itself.
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solution of an equation
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A value of a variable that makes a true statement when substituted into the equation. The process of finding the solution to an equation is called solving the equation.
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term
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A constant or the product of a constant and one or more variables.
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Key Concepts
2.1 - Use the Language of Algebra
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Operation
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Notation
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Say:
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The result is…
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Addition
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a + b
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a plus b
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The sum of a and b
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Multiplication
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a • b, (a)(b), (a)b, a(b)
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a times b
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The product of a and b
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Subtraction
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a - b
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a minus b
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The difference of a and b
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Division
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a ÷ b, a / b, \(\dfrac{a}{b}\), \(b \overline{)a}\)
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a divided by b
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The quotient of a and b
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Equality Symbol
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a = b is read as a is equal to b
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The symbol = is called the equal sign.
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Inequality
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a < b is read a is less than b
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a is to the left of b on the number line:
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a > b is read a is greater than b
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a is to the right of b on the number line:
Table 2.77
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Algebraic Notation
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Say
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a = b
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a is equal to b
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a ≠ b
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a is not equal to b
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a < b
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a is less than b
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a > b
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a is greater than b
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a ≤ b
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a is less than or equal to b
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a ≥ b
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a is greater than or equal to b
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Exponential Notation
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For any expression a n is a factor multiplied by itself n times, if n is a positive integer.
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a
n
means multiply n factors of a
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The expression of a
n
is read a to the n
th
power
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Order of Operations
: When simplifying mathematical expressions perform the operations in the following order:
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Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
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Exponents: Simplify all expressions with exponents.
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Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.
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Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.
2.2 - Evaluate, Simplify, and Translate Expressions
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Identify like terms.
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Rearrange the expression so like terms are together.
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Add the coefficients of the like terms
2.3 - Solving Equations Using the Subtraction and Addition Properties of Equality
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Determine whether a number is a solution to an equation.
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Substitute the number for the variable in the equation.
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Simplify the expressions on both sides of the equation.
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Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.
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Subtraction Property of Equality
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For any numbers a, b, and c, if a = b, then a - c = b - c.
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Solve an equation using the Subtraction Property of Equality.
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Use the Subtraction Property of Equality to isolate the variable.
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Simplify the expressions on both sides of the equation.
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Check the solution.
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Addition Property of Equality
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For any numbers a, b, and c, if a = b, then a + c = b + c.
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Solve an equation using the Addition Property of Equality.
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Use the Addition Property of Equality to isolate the variable.
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Simplify the expressions on both sides of the equation.
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Check the solution.
2.4 - Find Multiples and Factors
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Divisibility Tests
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A number is divisible by
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2
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if the last digit is 0, 2, 4, 6, or 8
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3
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if the sum of the digits is divisible by 3
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5
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if the last digit is 5 or 0
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6
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if divisible by both 2 and 3
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10
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if the last digit is 0
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Factors:
If a • b = m, then a and b are factors of m, and m is the product of a and b.
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Find all the factors of a counting number.
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Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor.
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If the quotient is a counting number, the divisor and quotient are a pair of factors.
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If the quotient is not a counting number, the divisor is not a factor.
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List all the factor pairs.
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Write all the factors in order from smallest to largest.
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Determine if a number is prime
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Test each of the primes, in order, to see if it is a factor of the number.
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Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found.
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If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.
2.5 - Prime Factorization and the Least Common Multiple
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Find the prime factorization of a composite number using the tree method
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Find any factor pair of the given number, and use these numbers to create two branches.
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If a factor is prime, that branch is complete. Circle the prime.
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If a factor is not prime, write it as the product of a factor pair and continue the process.
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Write the composite number as the product of all the circled primes.
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Find the prime factorization of a composite number using the ladder method
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Divide the number by the smallest prime.
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Continue dividing by that prime until it no longer divides evenly.
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Divide by the next prime until it no longer divides evenly.
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Continue until the quotient is a prime.
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Write the composite number as the product of all the primes on the sides and top of the ladder.
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Find the LCM by listing multiples
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List the first several multiples of each number.
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Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number.
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Look for the smallest number that is common to both lists.
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This number is the LCM.
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Find the LCM using the prime factors method
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Find the prime factorization of each number.
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Write each number as a product of primes, matching primes vertically when possible.
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Bring down the primes in each column.
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Multiply the factors to get the LCM.
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