Slide Copyright © 2009 Pearson Education, Inc. 3.1 Order of Operations

Slide Copyright © 2009 Pearson Education, Inc. Definitions Algebra: a generalized form of arithmetic. Variables: letters used to represent numbers Constant: symbol that represents a specific quantity Algebraic expression: a collection of variables, numbers, parentheses, and operation symbols. Examples:

Slide Copyright © 2009 Pearson Education, Inc. Order of Operations 1.First, perform all operations within parentheses or other grouping symbols (according to the following order). 2.Next, perform all exponential operations (that is, raising to powers or finding roots). 3.Next, perform all multiplication and divisions from left to right. 4.Finally, perform all additions and subtractions from left to right. Remember as: PEMDAS or Please Excuse My Dear Aunt Sally

Slide Copyright © 2009 Pearson Education, Inc. Example: Substituting for Two Variables Evaluate when x = 3 and y = 4. Solution: -4x 2 + 3xy – 5y 2 -4(3) 2 + 3(3)(4) – 5(4) 2 -4(9) + 36 – 5(16) –

Slide Copyright © 2009 Pearson Education, Inc. 3.2 Linear Equations in One Variable

Slide Copyright © 2009 Pearson Education, Inc. Definitions Terms are parts that are added or subtracted in an algebraic expression. Coefficient is the numerical part of a term. Like terms are terms that have the same variables with the same exponents on the variables. Unlike terms have different variables or different exponents on the variables.

Slide Copyright © 2009 Pearson Education, Inc. Properties of the Real Numbers Associative property of multiplication (ab)c = a(bc) Associative property of addition (a + b) + c = a + (b + c) Commutative property of multiplication ab = ba Commutative property of addition a + b = b + a Distributive propertya(b + c) = ab + ac

Slide Copyright © 2009 Pearson Education, Inc. Example: Combine Like Terms 8x + 4x = 3x y x =

Slide Copyright © 2009 Pearson Education, Inc. Think of a balanced scale…

Slide Copyright © 2009 Pearson Education, Inc. Solving Equations Addition Property of Equality If a = b, then a + c = b + c for all real numbers a, b, and c. Find the solution to the equation x - 9 = 24. Check: =

Slide Copyright © 2009 Pearson Education, Inc. Solving Equations continued Subtraction Property of Equality If a = b, then a - c = b - c for all real numbers a, b, and c. Find the solution to the equation x + 12 = 31. x = x = 19 Check: x + 12 = = 31 ? 31 = 31 true

Slide Copyright © 2009 Pearson Education, Inc. Solving Equations continued Multiplication Property of Equality If a = b, then a c = b c for all real numbers a, b, and c. Find the solution to the equation

Slide Copyright © 2009 Pearson Education, Inc. Solving Equations continued Division Property of Equality If a = b, then for all real numbers a, b, and c,. Find the solution to the equation 4x = 48.

Slide Copyright © 2009 Pearson Education, Inc. General Procedure for Solving Linear Equations If the equation contains fractions, multiply both sides of the equation by the lowest common denominator (or least common multiple). This step will eliminate all fractions from the equation. Use the distributive property to remove parentheses when necessary. Combine like terms on the same side of the equal sign when possible.

Slide Copyright © 2009 Pearson Education, Inc. General Procedure for Solving Linear Equations continued Use the addition or subtraction property to collect all terms with a variable on one side of the equal sign and all constants on the other side of the equal sign. It may be necessary to use the addition or subtraction property more than once. This process will eventually result in an equation of the form ax = b, where a and b are real numbers. Solve for the variable using the division or multiplication property. The result will be an answer in the form x = c, where c is a real number.

Slide Copyright © 2009 Pearson Education, Inc. Example: Solving Equations Solve 3x - 4 = 17. 3x - 4 = 17.

Slide Copyright © 2009 Pearson Education, Inc. Example: Solving Equations Solve 8x + 3 = 3(2x + 7). 8x + 3 = 3(2x + 7)

Slide Copyright © 2009 Pearson Education, Inc. Example: Solving Equations Solve False, the equation has no solution. The equation is inconsistent.

Slide Copyright © 2009 Pearson Education, Inc. Example: Solving Equations Solve True, 0 = 0 the solution is all real numbers.

Slide Copyright © 2009 Pearson Education, Inc. Proportions A proportion is a statement of equality between two ratios. Cross Multiplication If then ad = bc.

Slide Copyright © 2009 Pearson Education, Inc. To Solve Application Problems Using Proportions Represent the unknown quantity by a variable. Set up the proportion by listing the given ratio on the left-hand side of the equal sign and the unknown and other given quantity on the right- hand side of the equal sign. When setting up the right-hand side of the proportion, the same respective quantities should occupy the same respective positions on the left and right. For example, an acceptable proportion might be

Slide Copyright © 2009 Pearson Education, Inc. To Solve Application Problems Using Proportions continued Once the proportion is properly written, drop the units and use cross multiplication to solve the equation. Answer the question or questions asked using appropriate units.

Slide Copyright © 2009 Pearson Education, Inc. Example A 50 pound bag of fertilizer will cover an area of 15,000 ft 2. How many pounds are needed to cover an area of 226,000 ft 2 ? 754 pounds of fertilizer would be needed.

Slide Copyright © 2009 Pearson Education, Inc. 3.3 Formulas

Slide Copyright © 2009 Pearson Education, Inc. Definitions A formula is an equation that typically has a real-life application. To evaluate a formula, substitute the given value for their respective variables and then evaluate using the order of operations.

Slide Copyright © 2009 Pearson Education, Inc. Perimeter The formula for the perimeter of a rectangle is Perimeter = 2 length + 2 width or P = 2l + 2w. Use the formula to find the perimeter of a yard when l = 150 feet and w = 100 feet. P = 2l + 2w P = 2(150) + 2(100) P = P = 500 feet

Slide Copyright © 2009 Pearson Education, Inc. Example The formula for the volume of a cylinder is Use the formula to find the height of a cylinder with a radius of 6 inches and a volume of in 3. The height of the cylinder is 5 inches.

Slide Copyright © 2009 Pearson Education, Inc. Exponential Equations: Carbon Dating Carbon dating is used by scientists to find the age of fossils, bones, and other items. The formula used in carbon dating is If 15 mg of C 14 is present in an animal bone recently excavated, how many milligrams will be present in 4000 years?

Slide Copyright © 2009 Pearson Education, Inc. Exponential Equations: Carbon Dating continued In 4000 years, approximately 9.2 mg of the original 15 mg of C 14 will remain.

Slide Copyright © 2009 Pearson Education, Inc. Solving for a Variable in a Formula or Equation Solve the equation 3x + 8y  9 = 0 for y.

Slide Copyright © 2009 Pearson Education, Inc. Solve for b 2.

Slide Copyright © 2009 Pearson Education, Inc. 3.4 Applications of Linear Equations in One Variable

Slide Copyright © 2009 Pearson Education, Inc. Translating Words to Expressions 2x2x Twice a number x – 8 A number decreased by 8 x – 4Four less than a number x + 5A number increased by 5 x + 10Ten more than a number Mathematical Expression Phrase

Slide Copyright © 2009 Pearson Education, Inc. Translating Words to Expressions Five less than 7 times a number x – 6 The difference between a number and 6 2 – x 2 decreased by a number 4x4xFour times a number Mathematical Expression Phrase 7x – 5

Slide Copyright © 2009 Pearson Education, Inc. Translating Words to Expressions 2x + 11 Eleven more than twice a number 9x – 5 Nine times a number, decreased by 5 6x + 4 The sum of 6 times a number and 4 Mathematical Expression Phrase

Slide Copyright © 2009 Pearson Education, Inc. Translating Words to Expressions 2x  3 = 8 Twice a number, decreased by 3 is 8. x – 3 = 4 Three less than a number is 4 x + 7 = 12 Seven more than a number is 12 Mathematical Equation Phrase x  15 = 9x A number decreased by 15 is 9 times the number

Slide Copyright © 2009 Pearson Education, Inc. To Solve a Word Problem Read the problem carefully at least twice to be sure that you understand it. If possible, draw a sketch to help visualize the problem. Determine which quantity you are being asked to find. Choose a letter to represent this unknown quantity. Write down exactly what this letter represents. Write the word problem as an equation. Solve the equation for the unknown quantity. Answer the question or questions asked. Check the solution.

Slide Copyright © 2009 Pearson Education, Inc. Example The bill (parts and labor) for the repairs of a car was $ The cost of the parts was $339. The cost of the labor was $45 per hour. How many hours were billed?

Slide Copyright © 2009 Pearson Education, Inc. Example Sandra Cone wants to fence in a rectangular region in her backyard for her lambs. She only has 184 feet of fencing to use for the perimeter of the region. What should the dimensions of the region be if she wants the length to be 8 feet greater than the width?

Slide Copyright © 2009 Pearson Education, Inc. continued, 184 feet of fencing, length 8 feet longer than width Let x = width of region Let x + 8 = length P = 2l + 2w x + 8 x