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1 Lesson Equations

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2 Lesson Equations California Standard: Algebra and Functions 1.1 Write and solve one-step linear equations in one variable. What it means for you: Youll learn about equations and how theyre related to expressions. Key Words: equation guess and check solve

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3 Youve done plenty of work with expressions now its time to tackle equations. Lesson Equations Theyre quite similar both involve variables, numbers, and combinations of the usual +, –, ×, and ÷ operations.

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4 An Equation Tells You That Two Things are Equal An equation tells you that something is equal to something else. Writing an equation is similar to writing an expression. It may involve a real-life problem, and you may have to use variables to represent unknown values. Lesson Equations These are all examples of equations: 9 – 3 = = 20 4 y + x = x – 7

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5 Example 1 Richard spent $3.50 on fruit and $7.26 on vegetables. He then spent $ x on meat, making a total of exactly $20. Write an equation to represent this. Solution follows… Lesson Equations Solution Richard spent x dollars, which is equal to $20. Writing this as an equation, x = 20. This can be simplified to x = 20.

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6 Example 2 Edward and Bianca each wrote a story in English class. Biancas story has 31 words more than Edwards. Write an equation to represent this. Solution follows… Lesson Equations Solution Create variables call the number of words in Biancas story b and the number of words in Edwards story e. Although both b and e are unknown amounts, the question says that b is 31 greater than e. Both equations show that b is 31 greater than e. So, a possible equation is e + 31 = b. You could also have written b – 31 = e.

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7 Guided Practice Solution follows… Lesson Write equations to represent the situations described in Exercises 1–3. 1.Jose scored 10 percent more than William on a class test. 2.Theresa owns two pairs of shoes more than Kimberly. 3.Richard and Tim each bake cakes. Richard notices that Tims cake is three times as heavy as his. Equations Using j for Joses percent, and w for Williams: either j – 10 = w or w + 10 = j Using t for the number of pairs of shoes owned by Theresa, and k for the number owned by Kimberly: either t – 2 = k or k + 2 = t Using r for the weight of Richards cake and t for Tims: t = 3 r

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8 Guided Practice Solution follows… Lesson Write equations to represent the situations described in Exercises 4–6. 4.Isaac goes to the county fair. Because he has been saving up, he has seven times as much money as his sister to spend on rides. 5.Lauras dad runs a newsstand that sells bottles of water. Laura knows that five small bottles of water contain the same amount as two large bottles. 6.Monica has a candy bar that is a third the size of Julios. Equations Using i for Isaacs money and s for his sisters: i = 7 s Using s for the volume of a small bottle and b for a big: 5 s = 2 b Using m for the size of Monicas bar and j for the size of Julios: m = j ÷ 3

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9 Equations Can Be Used to Find the Value of Variables Equations like y + 6 = 16 tell you enough information to find out y. One way to find y is to guess at it again and again until you find the right answer. Sometimes this is called the guess and check method. Lesson Equations

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10 Example 3 Find the value of y if y + 6 = 16. Solution follows… Lesson Equations Solution First you need to guess a value for y. Then you can use it in the equation and check if it makes the equation true. Try y = 5: = 16 is not true. In fact, = 11, which is too small. Try y = 12: = 16 is also not true. In fact, = 18, which is too big. Try y = 10: = 16, which is true. So y does equal 10. This means the solution to the equation is y = 10.

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11 Example 4 Julia uses the expression 7 h to find out how many dollars she earned doing yard work. h represents the number of hours she worked. If Julia earned $56, then how many hours did she work? Solution follows… Lesson Equations Solution An equation for the situation is 7 h = 56. Try guessing different values of h, then see if theyre right. h 7h7h 7 h = 56? 749No 1070No 856Yes If Julia worked for 7 hours, she would have earned only $49, which is too little. If Julia worked for 10 hours, she would have earned $70, which is too much. Julia must have worked for 8 hours because when h = 8, the equation 7 h = 56 is true.

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12 Guided Practice Solution follows… Lesson Use guess and check to solve the equations in Exercises 7– x + 14 = z + 16 = a – 12 = z ÷ 7 = 6 Equations x = 16 z = 11 a = 9 z = 14

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13 Guided Practice Solution follows… Lesson Luis uses the expression $3 × m to work out how much money he spends calling his relatives in Europe for m minutes. His last phone call cost $60. Write an equation to represent this situation and then find out how long he spoke for. Equations 3 m = 60, Luis spoke for 20 minutes

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14 Guided Practice Solution follows… Lesson Pedro works out how much he must pay toward a food bill by using the formula d – $3.24, where d is the cost of the entire bill. He ends up paying $2.97. Write an equation for this situation and then find out how much the food bill must have been. 13.What value of y satisfies the equation 111 = 19 y + 16? Equations d – 3.24 = 2.97, the food bill was $6.21 E.g. Try y = 10, 111 = 19 × = = 206 – too big Try y = 5, 111 = 19 × = = 111, so y = 5.

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15 Independent Practice Solution follows… Lesson Equations Use guess and check to solve the equations in Exercises 1–4. 1. a + 20 = b + 9 = c – 22 = d ÷ 8 = 6 a = 37 b = 2 c = 6 d = 4

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16 Independent Practice Solution follows… Lesson Equations 5.Brandon has $ t in his savings jar. He puts in an extra $12. If the total amount in the jar is n, write an equation linking n and t. 6.Solve the equation you wrote for Exercise 5 when n = Cody is playing an old video game. In it you get 400 points every time you collect a ring and 800 points every time you collect an egg. Write an equation about p, the total number of points you get for collecting r rings and e eggs. t + 12 = n t = 36 p = 400 r e

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17 Independent Practice Solution follows… Lesson Equations 8.Guadalupe has started going to dance class. Each class costs $3 and she spent $23.50 on dance shoes. Including the shoes, she has spent $71.50 in total. Write an equation for this and solve it to find out how many lessons Guadalupe has been to. 9.It takes 2 eggs to make 20 cookies, and 4 eggs to make 40 cookies. Which equation best describes the number of eggs, x, that it takes to make y cookies? If n is the number of lessons Guadalupe has been to, = n. Guadalupe has been to 16 lessons. y + x = 10 y = 10 xy – x = 10 x = 10 y

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18 Round Up Equations look like expressions, but they contain a = which tells you that something is equal to something else. Dont worry if youre not 100% comfortable with equations yet all of the Lessons in this Section are about equations, so youll have many chances to practice. Lesson Equations

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