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Unit II Equations

Unit II Equations. Solving Equations. What is an equation?. A mathematical statement that two expressions are equal. When solving any equation you want to ISOLATE THE VARIABLE. A variable is isolated when it appears by itself on one side of an equation

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Unit II Equations

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  1. Unit II Equations

  2. Solving Equations What is an equation? A mathematical statement that two expressions are equal When solving any equation you want to ISOLATE THE VARIABLE. A variable is isolated when it appears by itself on one side of an equation and not all on the other side. Isolate a variable by using inverse operations, which “undo” operations on the variable. An equation is like a balanced scale. To keep the balance, you must perform the same operation on both sides. Inverse Operations Add x Subtract x Multiply x Divide x

  3. Solving Equations • Algebraic equations are mathematical statements that two expressions are equal to each other. • When solving for a variable, you must ISOLATE the variable, meaning get the variable alone. • Check to see if you can simplify either side. (combine like terms, distribute etc.) • Get all of the variables to one side. (Hint: Try moving the “smaller” variable term. ) • Move all of the numbers to the other side by performing the opposite operations. • CHECKyour solution to make sure it works. (Replace the variable with the number you got as the solution.)

  4. Solving Equations using Addition & Subtraction Since 10 is subtracted from x, add 10 to both sides to undo the subtraction. + 10 +10 Since 7 is added to x, subtract 7 from both sides to undo the addition. - 7 -7 Try these:

  5. Solving Equations using Multiplication & Division (-5) (-5) Since x is divided by -5, multiply both sides by -5 to undo the division. Since x is multiplied by 12, divide both sides by 12 to undo the multiplication. ____ ____ 12 12 Try these:

  6. Solving Multi-Step Equations Do any adding or subtracting first! Remember, you want to ISOLATE the variable. -6 -6 Multiplication or Division comes next! ____ ____ -2 -2 Simplify each side of the equation if possible, combine like terms. Add or subtract, remembering to move constants AWAY from the Variable. Multiply or Divide to get the variable alone.

  7. Try these: 1) 7) 2) 8) 3) 9) 4) 10) 5) 11) all real #s 6) 12)

  8. When solving an EQUATION: • GOAL: ISOLATE the variable (get the variable alone). • Check to see if you can simplify either side. (distribute, combine like terms etc.) • Get all of the variables to one side (using inverse operations). • Move all of the numbers to the other side (using inverse operations) • CHECKyour solution to make sure it works. (Replace the variable with the number you get as the solution.)

  9. Applications • Stephen belongs to a movie club in which he pays an annual fee of $39.95 and then rents DVDs for $0.99 each. In one year, Stephen spent $55.79. Write and solve an equation to find how many DVDs he rented. • Equation:39.95 + 0.99 d= 55.79 Answer: 16 DVDs • Maggie’s brother is three years younger than twice her age. The sum of their ages is 24. How old is Maggie? • Answer: 9 years old • The sum of two consecutive whole numbers is 57. What are the two numbers? • Answer: 28 and 29 • The height of an ostrich is 20 inches more than 4 times the height of a kiwi. Write and solve an equation to find the height of a kiwi. • Equation: 4k+20 = 108 Answer: 22 inches

  10. Solving Absolute Value Equations Remember that the absolute value of a number is that number’s distance from zero on a number line. For any nonzero absolute value, there are exactly two numbers with that absolute value. For examples, both 5 and -5 have an absolute value of 5. To write this statement using algebra, you would write . This equation asks, “What values of x have an absolute value of 5?” The solutions are 5 and -5. Ex. A Since is multiplied by 4, divide both sides by 4 to undo the multiplication. _____ ___ 4 4 Rewrite the equation as two cases. Since 2 is added to x, subtract 2 from both sides of the equation. The solutions are 4 and -8. * Remember, absolute value cannot be negative! Case 1 -2 -2 Case 2 -2 -2

  11. Solving Proportions Proportion: a statement that two ratios are equal * Cross multiply to solve Ex. A Ex. B

  12. Absolute Value & Proportion Practice 1) 6) 2) 7) 3) 8) 4) 9) 5) 10)

  13. Percents A percent is a ratio that compared a number to 100. For example, 25% = Fraction equivalent of a %: Decimal equivalent of a %: Finding the Part Ex. A Find 50% of 20. Use the percent proportion. Let x represent the percent. Find the cross products. Since x is multiplied by 24, divide both sides by 24 to undo the multiplication.

  14. Percent Practice • 1) What % of 60 is 15? 25 7) Kate found a new dress at the mall. The • price tag reads $90. The sign above the rack • of dresses says that all items are 40% off. • 2) 440 is what % of 400? 110%How much will Kate pay for the dress? • $54 • 8) On average, sloths spend 16.5 hours per • 3) 40% of what number is 14? 35 day sleeping. What percent of the day do • sloths spend sleeping? Round your answer • to the nearest percent. • 69% • 4) Find 105% of 72. 75.6 • 9) A can of ice tea contains 4% of the • recommended daily allowance of sodium. • The recommended daily allowance is 2500 mg • 5) 36 is 90% of what number? 40How many milligrams are in the can of ice tea? • 100 mg • 10) A newspaper reported that 42% of • 5 is what percent of 50? 10 Registered voters, voted in the election. If 12,000 people voted, how many registered voters are there? • 28,572

  15. Unit II: Application Problems • A taxi company charges $2.10 plus $0.80 per mile. Carmen paid a fare of $11.70. Write and solve an equation to find the number of miles she traveled. • $2.10 + .80m=$11.70 m = 12 miles • 2) On the first day of the year, David had $700 in his savings account and started spending $35 a week. His brother John had $450 and started saving $15 a week. After how many weeks will the brothers have the same amount? What will that amount be? • 5 weeks $525 • Peter earns $32,000 per year plus a 2.5% commission on his jewelry sales. Find Peter’s total salary for the year when his sales are valued at $420,000. • $42,500 • A volunteer at the zoo is responsible for feeding the animals in 15 exhibits in the reptile house. This represents 20% of the total exhibits in the reptile house. How many exhibits are in the reptile house? • 75 exhibits

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