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Algebra Basics and Linear Equations

Learn the fundamental concepts of algebra, including linear equations, solving methods, types of linear equations, applications such as mixture problems, and variation techniques. Explore ratio, proportion, quadratic equations, and special products, all explained with examples and practical problems.

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Algebra Basics and Linear Equations

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  1. Chapter 7: Basic Concepts of Algebra

  2. 7.1 Linear Equations • An equation in the variable x is linear if it can be written in the form Ax + B = C where A,B,C are real numbers and A is not 0.

  3. 7.1 Helpful properties • Addition property: A = B and A + C = B + C are equivalent (same solutions). • Multiplication property: A = B and AC = BC Are equivalent as long as C is not 0.

  4. 7.1 Solving a linear equation • Clear fractions • Simplify each side separately • Isolate the variable terms on one side • Transform so that the coefficient of the variable is 1 • Check your solution

  5. 7.1 Kinds of Linear equations • Conditional: finite number of solutions Ex. 2x = 4 • Contradiction: no solutions Ex. 2x + 1 = 2x +5 • Identity: true for any number Ex. 2x + 2 = 2(x + 1)

  6. 7.2 Applications of Linear Equations

  7. 7.2 Examples If a quotient of a number and 6 is added to twice the number, the result is 8 less than the number

  8. 7.2 Solving an Applied Problem • Read the problem carefully • Assign a variable to the unknown value, and write down any other unknowns in terms of this variable. Use tables, diagrams, etc. • Write equation using the variable. • Solve the equation. • State the answer in words relative to the context. Is it reasonable? • Check the answer in the words of the original problem.

  9. 7.2 Coin Mixture Dave collects US gold coins. He has a collection of 80 coins. Some are $10 coins and some are $20 coins. If the face value of the coins is $1060, how many of each denomination does he have?

  10. 7.2 Alcohol Mixture How many liters of a 20% alcohol solution must be mixed with 40 liters of a 50% solution to get a 40% solution?

  11. Distance, Rate and Time

  12. 7.2 #62 Time Traveled on a Visit Steve leaves Nashville to visit his cousin Dave in Napa, 80 miles away. He travels at an average speed of 50 miles per hour. One half-hour later Dave leaves to visit Steve, traveling at an average speed of 60 miles per hour. How long after Dave leaves will they meet?

  13. #57 Travel Times of Trains A train leaves Little Rock, Arkansas, and travels north at 85 km/hr. Another train leaves at the same time and travels south at 95 km/hr. How long will it take before they are 315 km apart?

  14. 7.3 Ratio, Proportion and Variation

  15. 7.3 Proportion and Cross Products

  16. #26: Proper Dosages A nurse is asked to administer 200 milligrams of fluconazole to a patient. There is a stock solution that provides 40 milligrams per milliliter. How much of the solution should be given to the patient?

  17. Golden Ratio/Divine Proportion Fibonacci sequence: 1, 1, 2, 3, 5, 8, … What happens to ratios of successive Fibonacci numbers? Golden ratio is 1.618…

  18. Fibonacci Squares

  19. 7.3 Direct Variation

  20. #44 Paddleboat Voyage According to Guinness World Records, the longest recorded voyage in a paddleboat is 2226 miles in 103 days by the foot power of two boaters down the Mississippi River. Assuming a constant rate, how far would they have gone if they had traveled 120 days? (Distance varies directly as time)

  21. 7.3 Direct Variation as a Power

  22. #51 Falling Body For a body falling freely from rest, the distance the body falls varies directly as the square of the time. If an object is dropped from the top of a tower 400 feet high and hits the ground in 5 seconds, how far did it fall in the first 3 seconds?

  23. 7.3 Inverse Variation

  24. Joint and Combined Variation If one variable varies as the product of several other variables, it is said to be in joint variation. Combined variation involves combinations of direct and inverse variation.

  25. #53 Skidding Car The force needed to keep a car from skidding on a curve varies inversely as the radius of the curve and jointly as the weight of the car and the square of the speed. If 242 pounds of force keep a 2000 pound car from skidding on a curve of radius 500 feet at 30 miles per hour, what force would keep the same car from skidding on a curve of radius 750 feet at 50 miles per hour?

  26. Properties of Exponents 7.5

  27. Power Rules for Exponents

  28. Special Products of Polynomials 7.6

  29. Quadratic Equations 7.7 • An equation of the form ax2 + bx + c = 0 where a,b,c are real numbers with a not equal to 0, is a quadratic equation. • Zero factor property (helpful for factoring): If AB = 0, then A = 0 or B = 0 or both

  30. Quadratic Formula

  31. Applications 7.7 The Toronto Dominion Centre in Winnipeg, Manitoba is 407 feet high. Suppose that a ball is projected upward from the top of the centre and its position s in feet above the ground is given by the equation s = -16t2 + 75t + 407, where t is the number of seconds elapsed. How long will it take for the ball to reach a height of 450 feet?

  32. 7.7 #68, page 355 • A club swimming pool is 30 feet wide by 40 feet long. The club members want a border in a strip of uniform width around the pool. They have enough material for 296 square feet. How wide can the strip be?

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