Evaluating algebraic expressions with substitution
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This document defines variables, constants, expressions and provides examples of evaluating expressions for given values of variables. It defines a variable as a quantity that can change and constants as quantities that do not change. Expressions are defined as mathematical phrases involving variables, constants and operations. Several examples are given of evaluating expressions when given values for variables like x, m, p, c, z.
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The document defines key terms used in algebraic expressions:
1) A variable represents an unknown value and can be letters or symbols like "B" in the expression "12 + B".
2) An algebraic expression uses variables with numbers and operations like "a + 2" or "3m + 6n - 6".
3) A coefficient is the number multiplied by a variable, like 6 is the coefficient of m in the expression "6m + 5".
4) A term refers to a number, variable, or their combination using multiplication or division, like "a" and "2" are terms in "a + 2".
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The document defines key terms used in algebraic expressions:
1) A variable represents an unknown value and can be letters or symbols like "B" in the expression "12 + B".
2) An algebraic expression uses variables with numbers and operations like "a + 2" or "3m + 6n - 6".
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3) Multiplication of rational numbers by multiplying numerators and denominators. Properties are closure, commutativity, associativity, identity of 1, and annihilation by 0.
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- Unlike terms cannot be combined.
- To add algebraic expressions, combine like terms by adding the coefficients.
- To subtract expressions, change the sign of the second expression and then add as if adding.
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The document provides steps for multiplying rational numbers:
1) Write rational numbers in the form a/b ∙ c/d
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1. When dividing a decimal by a whole number, place the decimal point in the quotient above the decimal point in the dividend. Divide as usual.
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The document provides an introduction to integers including:
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- Rules for comparing and ordering integers
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- Key concepts and rules for adding and subtracting integers including keeping the sign of the first integer the same and changing the operation and second integer's sign for subtraction
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- A method for proving that subtracting a negative number is the same as adding a positive number
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1) Determining whether the absolute value is greater than or less than the variable, which indicates a disjunction or conjunction graph.
2) Solving the inequality for both possibilities of the expression inside the absolute value being positive or negative.
3) Combining the solutions from both possibilities using the appropriate inequality symbol (>, <, etc.) to obtain the final solution set.
Multiplying Monomials
1. The document discusses multiplying and simplifying monomial expressions. It defines monomials and explains the rules for multiplying them, including adding exponents when multiplying like terms and multiplying all exponents when an expression is raised to a power.
2. The document also discusses multiplying monomials and polynomials using the distributive property and applying the same exponent rules. It provides examples of multiplying and simplifying expressions involving monomials and polynomials.
3. The learning objectives are to multiply monomials, simplify expressions with monomials, and to multiply a monomial and a polynomial.
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This document discusses properties of operations on rational numbers. It covers:
1) Addition of rational numbers, including having the same or different denominators. Properties include closure, commutativity, and additive identity.
2) Subtraction of rational numbers and its properties, noting the difference property and lack of an identity element.
3) Multiplication of rational numbers by multiplying numerators and denominators. Properties are closure, commutativity, associativity, identity of 1, and annihilation by 0.
4) Distributive property relating multiplication and addition/subtraction of rational numbers.
Operations with integers
Integers include all whole numbers and their opposites, and zero. The key rules for operations with integers are:
1) Adding integers of the same sign yields a positive sum, while those of opposite signs subtracts and uses the sign of the larger number.
2) Multiplying and dividing integers follows sign rules - same signs yield a positive result, different signs a negative one.
3) To subtract integers, change the operation to addition by changing the subtraction sign to addition and changing the sign of the number being subtracted.
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This document discusses simplifying rational expressions by dividing out common factors, factoring numerators and denominators, and dividing common factors. It provides examples of simplifying various rational expressions step-by-step and explains how to identify excluded values that would make denominators equal to zero.
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- Like terms are algebraic expressions with the same variables and exponents.
- Unlike terms cannot be combined.
- To add algebraic expressions, combine like terms by adding the coefficients.
- To subtract expressions, change the sign of the second expression and then add as if adding.
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1) Find equivalent fractions with a common denominator
2) Add the numerators and use the sum as the new numerator
3) Keep the common denominator as the denominator
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Worked examples demonstrate applying the steps to add several pairs of fractions.
Adding and subtracting polynomials
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- 2. 5w + 18 Variable: Is a quantity that can change or vary. Mathematicians use variables (letters) to represent numbers. Constant: A quantity that does not change. Expression: Is a mathematical phrase involving constants, variables, and operation symbols.
- 3. Evaluate each expression for x=1, 2, and 3. x 1 2 3 x+5
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- 8. x = 3 y = 12 z = 8 4z + 8 – 6 6x – 9 ÷ 3 15 + 9x ÷ 3 7z ÷ 4 + 5x 2 ÷ 3x y