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Algebraic Expressions and Equations: Solving Linear Equations

Learn about algebra, order of operations, like terms, and solving linear equations in this comprehensive guide. Practice evaluating expressions, substituting variables, and applying properties of equality.

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Algebraic Expressions and Equations: Solving Linear Equations

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  1. SEVENTH EDITION and EXPANDED SEVENTH EDITION

  2. Chapter 6 Algebra, Graphs and Functions

  3. 6.1 Order of Operations

  4. Definitions • Algebra: a generalized form of arithmetic. • Variables: used to represent numbers • Algebraic expression: a collection of variables, numbers, parentheses, and operation symbols. • Examples:

  5. 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.

  6. Example: Evaluating an Expression • Evaluate the expression x2 + 4x + 5 for x = 3. • Solution: x2 + 4x + 5 = 32 + 4(3) + 5 = 9 + 12 + 5 = 26

  7. Example: Substituting for Two Variables • Evaluate when x = 3 and y = 4. • Solution:

  8. 6.2 Linear Equations in One Variable

  9. Definitions • 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.

  10. a(b + c) = ab + ac Distributive property a + b = b + a Commutative property of addition ab = ba Commutative property of multiplication (a + b) + c = a + (b + c) Associative property of addition (ab)c = a(bc) Associative property of multiplication Properties of the Real Numbers

  11. 8x + 4x = (8 + 4)x = 12x 5y 6y = (5  6)y = y x + 15  5x + 9 = (1 5)x + (15+9) = 4x + 24 3x + 2 + 6y  4 + 7x = (3 + 7)x + 6y + (2  4) = 10x + 6y  2 Example: Combine Like Terms

  12. 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. x  9 + 9 = 24 + 9 x = 33 Check: x 9 = 24 33  9 = 24 ? 24 = 24 true Solving Equations

  13. 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 + 12  12 = 31  12 x = 19 Check: x+ 12 = 31 19 + 12 = 31 ? 31 = 31 true Solving Equations continued

  14. Multiplication Property of Equality If a = b, then a• c = b • c for all real numbers a, b, and c, where c 0. Find the solution to the equation Solving Equations continued

  15. Division Property of Equality If a = b, then for all real numbers a, b, and c, c 0. Find the solution to the equation 4x = 48. Solving Equations continued

  16. 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.

  17. 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.

  18. General Procedure for Solving Linear Equations continued • Solve for the variable using the division or multiplication property. This will result in an answer in the form x = c, where c is a real number.

  19. Example: Solving Equations • Solve 3x 4 = 17.

  20. Solve 21 = 6 + 3(x + 2)

  21. Solve 8x + 3 = 6x + 21

  22. Solve 6(x 2) + 2x + 3 = 4(2x  3) + 2 • False, the equation has no solution. The equation is inconsistent.

  23. Solve 4(x + 1)  6(x + 2) = 2(x + 4) • True, 0 = 0 the solution is all real numbers.

  24. Proportions • A proportion is a statement of equality between two ratios. • Cross Multiplication • If then ad = bc, b 0, d 0.

  25. 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.

  26. To Solve Application Problems Using Proportions continued • For example, an acceptable proportion might be • Once the proportion is properly written, drop the units and use cross multiplication to solve the equation. • Answer the question or questions asked.

  27. A 50 pound bag of fertilizer will cover an area of 15,000 ft2. How many pounds are needed to cover an area of 226,000 ft2? 754 pounds of fertilizer would be needed. Example

  28. 6.3 Formulas

  29. Definitions • A formula is an equation that typically has a real-life application.

  30. 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 = 300 + 200 P = 500 feet

  31. Example • The formula for the volume of a cylinder is V = r2h. Use the formula to find the height of a cylinder with a radius of 6 inches and a volume of 565.49 in3. • The height of the cylinder is 5 inches.

  32. 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 C14 is present in an animal bone recently excavated, how many milligrams will be present in 4000 years?

  33. Exponential Equations: Carbon Dating continued • In 4000 years, approximately 9.2 mg of the original 15 mg of C14 will remain.

  34. Solving for a Variable in a Formula or Equation • Solve the equation 3x + 8y 9 = 0 for y.

  35. Solve for b2.

  36. 6.4 Applications of Linear Equations in One Variable

  37. Phrase Mathematical Expression Ten more than a number x + 10 Five less than a number x 5 Twice a number 2x Eight more than twice a number 2x + 8 The sum of three times a number decreased by 9. 3x 9 Translating Words to Expressions

  38. 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.

  39. Example The bill (parts and labor) for the repairs of a car was $496.50. The cost of the parts was $339. The cost of the labor was $45 per hour. How many hours were billed? • Let h = the number of hours billed • Cost of parts + labor = total amount 339 + 45h = 496.50

  40. Example continued • The car was worked on for 3.5 hours.

  41. 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?

  42. x x + 8 continued, 184 feet of fencing, length 8 feet longer than width • Let x = width of region • Let x + 8 = length • P = 2l + 2w The width of the region is 42 feet and the length is 50 feet.

  43. 6.5 Variation

  44. Direct Variation • Variation is an equation that relates one variable to one or more other variables. • In direct variation, the values of the two related variables increase or decrease together. • If a variable y varies directly with a variable x, then y = kx where k is the constant of proportionality(or the variation constant).

  45. Example • The amount of interest earned on an investment, I, varies directly as the interest rate, r. If the interest earned is $50 when the interest rate is 5%, find the amount of interest earned when the interest rate is 7%. • I = rx $50 = 0.05x 1000 = x

  46. Example continued • x = 1000, r = 7% I = rx I = 0.07(1000) I = $70 • The amount of interest earned is $70.

  47. Inverse Variation • When two quantities vary inversely, as one quantity increases, the other quantity decreases, and vice versa. • If a variable y varies inversely with a variable, x, then y = k/x where k is the constant of proportionality.

  48. Suppose y varies inversely as x. If y = 12 when x = 18, find y when x = 21. Now substitute 216 for k in y = k/x and find y when x = 21. Example

  49. Joint Variation • One quantity may vary directly as the product of two or more other quantities. • The general form of a joint variation, where y, varies directly as x and z, is y = kxz where k is the constant of proportionality.

  50. Example • The area, A, of a triangle varies jointly as its base, b, and height, h. If the area of a triangle is 48 in2 when its base is 12 in. and its height is 8 in., find the area of a triangle whose base is 15 in. and whose height is 20 in. • A = kbh

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