2.S: Introduction to the Language of Algebra (Summary)
- Page ID
- 5965
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Key Terms
| coefficient | The constant that multiplies the variable(s) in a term. |
| composite number | A composite number is a counting number that is not prime. |
| divisibility | If a number m is a multiple of n, then we say that m is divisible by n. |
| equation | An equation is made up of two expressions connected by an equal sign. |
| evaluate | To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. |
| expression | An expression is a number, a variable, or a combination of numbers and variables and operation symbols. |
| least common multiple (LCM) | The smallest number that is a multiple of two numbers. |
| like terms | Terms that are either constants or have the same variables with the same exponents. |
| multiple of a number | A number is a multiple of n if it is the product of a counting number and n. |
| prime factorization | The product of prime numbers that equals the number. |
| prime number | A counting number greater than 1 whose only factors are 1 and itself. |
| solution of an equation | 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. |
| term | A constant or the product of a constant and one or more variables. |
Key Concepts
2.1 - Use the Language of Algebra
| Operation | Notation | Say: | The result is… |
|---|---|---|---|
| Addition | a + b | a plus b | The sum of a and b |
| Multiplication | a • b, (a)(b), (a)b, a(b) | a times b | The product of a and b |
| Subtraction | a - b | a minus b | The difference of a and b |
| Division | a ÷ b, a / b, \(\dfrac{a}{b}\), \(b \overline{)a}\) | a divided by b | The quotient of a and b |
- Equality Symbol
- a = b is read as a is equal to b
- The symbol = is called the equal sign.
- Inequality
- a < b is read a is less than b
- a is to the left of b on the number line:
- a > b is read a is greater than b
- a is to the right of b on the number line:
Table 2.77
| Algebraic Notation | Say |
|---|---|
| a = b | a is equal to b |
| a ≠ b | a is not equal to b |
| a < b | a is less than b |
| a > b | a is greater than b |
| a ≤ b | a is less than or equal to b |
| a ≥ b | a is greater than or equal to b |
- Exponential Notation
- For any expression a n is a factor multiplied by itself n times, if n is a positive integer.
- an means multiply n factors of a
- The expression of an is read a to the nth power
- Order of Operations: When simplifying mathematical expressions perform the operations in the following order:
- Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
- Exponents: Simplify all expressions with exponents.
- Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.
- 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
- Combine like terms.
- Identify like terms.
- Rearrange the expression so like terms are together.
- Add the coefficients of the like terms
2.3 - Solving Equations Using the Subtraction and Addition Properties of Equality
- Determine whether a number is a solution to an equation.
- Substitute the number for the variable in the equation.
- Simplify the expressions on both sides of the equation.
- 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.
- Subtraction Property of Equality
- For any numbers a, b, and c, if a = b, then a - c = b - c.
- Solve an equation using the Subtraction Property of Equality.
- Use the Subtraction Property of Equality to isolate the variable.
- Simplify the expressions on both sides of the equation.
- Check the solution.
- Addition Property of Equality
- For any numbers a, b, and c, if a = b, then a + c = b + c.
- Solve an equation using the Addition Property of Equality.
- Use the Addition Property of Equality to isolate the variable.
- Simplify the expressions on both sides of the equation.
- Check the solution.
2.4 - Find Multiples and Factors
| Divisibility Tests | |
|---|---|
| A number is divisible by | |
| 2 | if the last digit is 0, 2, 4, 6, or 8 |
| 3 | if the sum of the digits is divisible by 3 |
| 5 | if the last digit is 5 or 0 |
| 6 | if divisible by both 2 and 3 |
| 10 | if the last digit is 0 |
- Factors: If a • b = m, then a and b are factors of m, and m is the product of a and b.
- Find all the factors of a counting number.
- Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor.
- If the quotient is a counting number, the divisor and quotient are a pair of factors.
- If the quotient is not a counting number, the divisor is not a factor.
- List all the factor pairs.
- Write all the factors in order from smallest to largest.
- Determine if a number is prime.
- Test each of the primes, in order, to see if it is a factor of the number.
- Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found.
- 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
- Find the prime factorization of a composite number using the tree method.
- Find any factor pair of the given number, and use these numbers to create two branches.
- If a factor is prime, that branch is complete. Circle the prime.
- If a factor is not prime, write it as the product of a factor pair and continue the process.
- Write the composite number as the product of all the circled primes.
- Find the prime factorization of a composite number using the ladder method.
- Divide the number by the smallest prime.
- Continue dividing by that prime until it no longer divides evenly.
- Divide by the next prime until it no longer divides evenly.
- Continue until the quotient is a prime.
- Write the composite number as the product of all the primes on the sides and top of the ladder.
- Find the LCM by listing multiples
- List the first several multiples of each number.
- Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number.
- Look for the smallest number that is common to both lists.
- This number is the LCM.
- Find the LCM using the prime factors method.
- Find the prime factorization of each number.
- Write each number as a product of primes, matching primes vertically when possible.
- Bring down the primes in each column.
- Multiply the factors to get the LCM.