Ready To Go On? Skills Intervention 2-1 Solving Equations by Adding or Subtracting

Transcription

1 Ready To Go On? Skills Intervention 2-1 Solving Equations by Adding or Subtracting Find these vocabulary words in Lesson 2-1 and the Multilingual Glossary. Vocabulary equation solution of an equation Solve Equations by Using Addition Solve the equation. x What operation is between x and 43? x To isolate x, you should 43 to both sides of the equation. x Find the value of x. Check your solution by substituting the value of x into the original equation Does your solution make the equation true? Solve Equations by Using Subtraction Solve each equation. A. m 8 22 What operation is between m and 8? m 8 22 To isolate m, you should 8 from both sides of the equation. m Find the value of m. Check your solution. B. 3 4 h 3 8 What operation is between h and 3 8? 3 4 h Does your solution make the equation true? To isolate h, you should 3 from both sides of the equation. 8 To subtract fractions with different denominators, find a h What is the common denominator for 3 4 and 3 8? h Rewrite the fractions with the common denominator. h Check your solution Does your solution make the equation true? 22 Holt Algebra 1

2 Ready to Go On? Problem Solving Intervention 2-1 Solving Equations by Adding or Subtracting To find the solution to an equation, isolate the variable. To isolate the variable, use inverse, or opposite, operations to undo operations on the variable. When Jeremy began his 9th grade year in school he was 5 ft 4 in. tall. When he was measured for his graduation gown at the end of his 12th grade year, he was 6 ft 3 in. tall. Write and solve an equation to find how many inches Jeremy grew over 4 years. Understand the Problem 1. How tall was Jeremy at the beginning of his 9th grade year? 2. How tall was Jeremy at the end of his 12th grade year? Make a Plan 3. What do you need to determine? 4. How many total inches is 5 ft 4 in.? 5. How many total inches is 6 ft 3 in.? 6. Let h represent the number of inches Jeremy grew in 4 years. Write an equation to represent this situation. Height in Grade 9 The number of inches Jeremy grew Height in Grade 12 Solve 7. Solve the equation by isolating h. 8. The number of inches that Jeremy grew in 4 years is. h 75 Isolate h. h Look Back 9. Substitute the solution for h into the equation you wrote in Exercise 6. Height in Grade 9 The number of inches Jeremy grew Height in Grade Does your solution make the equation true? 23 Holt Algebra 1

3 Solving Equations by Using Multiplication Solve the equation and check your answer. x 18 4 What operation is between x and 4? x 18 To isolate x, you should both sides of the equation by 4. 4 x 4 (18) x Find the value of x. Check your solution by substituting the value of x into the original equation Does your solution make the equation true? Solving Equations by Using Division Solve each equation. Check your answers. A. 28 7n Ready to Go On? Skills Intervention 2-2 Solving Equations by Multiplying or Dividing What operation is between n and 7? 28 7n To isolate n, you should both sides of the equation by n n Find the value of n. Check your solution Does your solution make the equation true? B. 0.3d 4.2 What operation is between d and 0.3? 0.3d 4.2 To isolate d, you should both sides of the equation by d 4.2 To divide by a number with a decimal point, point to the right d Solve for d. Check your solution Does your solution make the equation true? place. the decimal 24 Holt Algebra 1

4 Ready to Go On? Problem Solving Intervention 2-2 Solving Equations by Multiplying or Dividing Multiplication and division are inverse, or opposite, operations. They undo each other. At the last football game of the season, there were 562 people. This was 2 of the 5 number of people who were at the first game of the season. Write an equation to find the number of people who were at the first game. Understand the Problem 1. How many people attended the last football game of the season? 2. Two-fifths of the number of people at the game is the same as the number of people at the Make a Plan 3. What do you need to determine? game. 4. What operation can be substituted for the word of? 5. Let n represent the number of people at the first football game. Write an equation to represent this situation. 2 5 of The number of people at the first game The number of people at the last game Solve 6. Solve the equation by isolating n. 7. The number of people that attended 2 5 n the first football game was. 2 5 n 5 2 Look Back n 8. Substitute the solution for n into the equation you wrote in Exercise of The number of people at the first game The number of people at the last game 9. Does your solution make the equation true? 25 Holt Algebra 1

5 Ready to Go On? Skills Intervention 2-3 Solving Two-Step Equations and Multi-Step Equations Solving Two-Step Equations Solve the equation. 5r To isolate the variable r, subtract from both sides. Since r is multiplied by, then both sides of the equation by. 5r r 5r 5 What is the value of r? Check: Substitute the value for r back into the original equation. Does your answer check? r Solving Two-Step Equations That Contain Fractions Solve the equation. 5 n To isolate the variable n, subtract from both sides. Since n is multiplied by equation by the reciprocal,. What is the value of n?, multiply both sides of the To check your answer, substitute the value for n back into the original equation. Does your answer check? Simplifying Before Solving Equations Solve the equation. 6(x 9) 3 To simplify, first multiply 6 with x and then with 9. To isolate the variable, x, add to both sides. Since x is multiplied by, divide both sides by. 5 n x n n 8 5 n x 6x 6 What is the value of x? To check your answer, substitute the value for x back into the original equation Does your answer check? x 26 Holt Algebra 1

6 Equations that contain more than one operation require more than one step to solve. Use inverse operations and work backward to undo each operation one step at a time. A fitness club offers membership for a joining fee of $99 and a monthly fee of $35. If Jenna has spent a total of $1009, how many months has Jenna been a member of the fitness club? Understand the Problem 1. How much is the joining fee for the fitness club? 2. How much does the fitness club cost each month? 3. How much total has Jenna spent on the fitness club? Make a Plan Ready to Go On? Problem Solving Intervention 2-3 Solving Two-Step Equations and Multi-Step Equations 4. What do you need to determine? 5. Let m represent the number of months that Jenna has been a member. Write an equation to represent the situation. Joining fee Monthly fee Number of months Jenna has been a member Total amount spent Solve m 6. Solve the equation by isolating m. 7. Jenna has been a member of the 99 m 1009 fitness club for months. m 35m Look Back m 8. Substitute the solution for m into the equation you wrote in Exercise 5. Joining fee Monthly fee Number of months Jenna has been a member Total amount spent 9. Does your solution make the equation true? 27 Holt Algebra 1

7 Ready to Go On? Skills Intervention 2-4 Solving Equations with Variables on Both Sides Find these vocabulary words in Lesson 2-4 and the Multilingual Glossary. Vocabulary identity contradiction Simplifying Each Side Before Solving Equations Solve the equation. 4(k 3) 3(3k 6) Use the 4 k 3 3 3k 6 Property to simplify this equation before you solve it. 4k k 18 To collect the variables on the right side, subtract from both sides. 4k 4k k 18 To collect the constants on the left side, subtract from both sides k 5k To isolate the variable, both sides by 5. k What is the value of k? Infinitely Many Solutions or No Solutions Solve the equation. 5 2x 3 2 5x 4 Use the 5 2x 3 2 5x 4 Property to simplify this equation before you solve it. 10x x 8 To collect the variables on the right side, subtract 10x 10x from both sides. 8 Complete the resulting equation. Is the resulting equation a true statement or a false statement? What is the other name for a false statement given in the lesson? How many values make this equation true? 28 Holt Algebra 1

8 Ready To Go On? Problem Solving Intervention 2-4 Solving Equations with Variables on Both Sides To solve equations with variables on both sides, begin by collecting the variable terms on one side of the equation. One electrician charges his customers a $60 service fee plus $35 per hour. Another electrician charges her customers $65 per hour. How many hours must the electricians work in order for the total cost of an electrician to be the same? What is the total cost? Understand the Problem 1. Describe how much the first electrician charges. 2. Describe how much the second electrician charges. Make a Plan 3. What do you need to determine? 4. Let h represent the number of hours that the electricians must work. Write an equation to represent the situation. Service fee for electrician #1 Hourly rate Number of hours electrician #1 works Hourly rate Number of hours electrician #2 works h h Solve 5. On which side should you collect the variable terms? 6. Solve the equation by isolating h. 60 h h h h 60 h 7. The number of hours that the electricians need to work for the total cost to be the same is hours. 8. What is the total cost for either of the electricians to work for this number of hours? 60 h h Look Back 9. Substitute the solution for h into the equation you wrote in Exercise 4. Service fee for electrician #1 Hourly rate Number of hours electrician #1 works Hourly rate Number of hours electrician #2 works 10. Does your solution make the equation true? 29 Holt Algebra 1

9 Ready To Go On? Skills Intervention 2-5 Solving for a Variable Find these vocabulary words in Lesson 2-5 and the Multilingual Glossary. Vocabulary formula literal equation Solving Literal Equations for a Variable A. Solve 7u v w 6 for w. Which side of the equation is w located? What constant is on the same side of the equation as w? Which operation is between the w and the constant? To isolate w, 6 to both sides of the equation. Complete: 7u v w 6 7u v w B. Solve j k 4(m 10) for m. Which side of the equation is m located? What property do you use to simplify the equation first? What is the resulting equation? j k m What constant is on the same side of the equation as 4m? Which operation is between the 4m and the constant? To isolate 4m, add to both sides of the equation. Complete to find the resulting equation. j k m j k 4m j k 4m j k m 30 Holt Algebra 1

10 Ready to Go On? Problem Solving Intervention 2-5 Solving for a Variable Rearrange a formula to isolate any variable by using inverse operations. Formulas are also called literal equations. Solve a literal equation by isolating one of the variables. The formula for the circumference of a circle is C 2r, where C is the circumference, r is the radius of the circle, and is the constant Solve the equation for r. If the circumference of a circle is 62.8 in., what is the radius of the circle? Understand the Problem 1. Which variable do you need to find? 2. What is the circumference of the given circle? What is? Make a Plan 3. To undo the multiplication between the 2 and r, both sides by. C 2r C 2r C r 4. To undo the multiplication between the and r, both sides by. C r 2 5. What is the formula for the radius of a circle, r? C r Solve 6. What value do you substitute in for C? and? 7. Simplify the equation and solve for r. r 2 8. The radius of the circle is. r Look Back 9. To check your solution, substitute the values into the original formula, C 2r. C 2 r Does your solution make the equation true? 31 Holt Algebra 1

11 Ready to Go On? Quiz 2-1 Solving Equations by Adding or Subtracting Solve each equation. 1. x k m h At the beginning of the week, Joshua had $356 in his savings account. At the end of the week he only had $123 in his savings account. Write and solve an equation to find the amount of money that Joshua spent during the week. 2-2 Solving Equations by Multiplying or Dividing Solve each equation. 6. x n y g There are 450 students who participate in extra-curricular activities at school. This is 5 of the total number of students enrolled in the school. Write and solve 8 an equation to find the number of students enrolled at the school. 2-3 Solving Two-Step and Multi-Step Equations Solve each equation r x Holt Algebra 1

12 Ready to Go On? Quiz continued 13. 8b 4 2b (f 7) A house painter charges each customer a $50 estimation fee and then $30 per hour. Write and solve an equation to find the number of hours the house painter worked if she earned $1250 at the last job. 2-4 Solving Equations with Variables on Both Sides Solve each equation x 5 4x (3x 6) 3(4x 3) 18. 4(2n 5) 7(2n 4) 19. 6(k 7) 6(k 7) 20. On the first day of the month, one investment began with $1568 and started losing $16 each day. Another investment began with $854 and started earning $26 each day. After how many days will the two investments have the same amount of money? What will that amount be? 2-5 Solving for a Variable 21. Solve 6m n p 7 for p. 22. Solve a b 4(c 5) for c. 23. The formula for the perimeter of a rectangle is P 2 2w. Solve the formula for w. If the perimeter of a rectangle is 64 in. and its length measures 14 in. what is the width of the rectangle? 33 Holt Algebra 1

13 Ready to Go On? Enrichment Diophantine Equations Diophantus (about ) is known to some as the father of algebra. He studied primarily the solutions of algebraic equations and the theory of numbers. One type of equation he studied has the form ax by c where a, b, and c are all integers and the solutions to the equation (x, y) are also integers. These types of equations are now known as Diophantine Equations. They can be quite difficult to solve and many times the only way to solve them is by guessing and checking. Solve each Diophantine Equation. Find at least one pair of positive integers for x and y that make the equation true. 1. 3x 4y x 3y 9 a. Solve the equation for y. a. Solve the equation for y. b. What number must x be divisible by? b. What number must x be divisible by? Why? Why? c. Find at least one solution. c. Find at least one solution. 3. x 2y x y x 19y x 7y x 11y x 4y x y x 5y 9 34 Holt Algebra 1

14 2B Find these vocabulary words in Lesson 2-6 and the Multilingual Glossary. Vocabulary Ready To Go On? Skills Intervention 2-6 Rates, Ratios, and Proportions ratio proportion cross products scale scale drawing rate unit rate scale model conversion factor Solving Proportions Solve each proportion. A m m 4 Complete the equation using cross products: 4 m Multiply. 11m To isolate m, both sides by. 11m What is the value of m? To check your answer, substitute the solution in for m m m Are the ratios equivalent? B. 6 r 3 6 r Complete the equation using cross products: 6 20 (r 3) To simplify the right side, use the Property r To simplify the left side, multiply. r 54 Subtract from both sides to isolate the variable r To solve for r, both sides by. What is the value of r? To check your answer, substitute the solution in for r. 6 r Are the ratios equivalent? 18r r 35 Holt Algebra 1

15 2B Ready to Go On? Problem Solving Intervention 2-6 Rates, Ratios, and Proportions A comparison of two quantities by division is a ratio. Two ratios that are equivalent is a proportion. The ratio of cats to dogs in the local animal shelter is 3:5. There are 60 dogs in the animal shelter. How many cats are in the animal shelter? Understand the Problem 1. What is the ratio of cats to dogs? 2. How many dogs are in the shelter? Make a Plan 3. What do you need to determine? 4. Complete the proportion: ratio of cats to dogs number of cats 5. If c represents the number of cats in the shelter, complete the proportion to Solve find the number of cats: 5 c 6. To solve the proportion, multiply both sides by. 7. Solve the equation for c. 8. The number of cats in the shelter is. 5 c 3 c 60 c Look Back 9. To check your solution, simplify the ratio of c to the number of dogs in the shelter. c Is this the same as the ratio of cats to dogs, 3:5? 36 Holt Algebra 1

16 2B Find these vocabulary words in Lesson 2-7 and the Multilingual Glossary. Vocabulary Ready to Go On? Skills Intervention 2-7 Applications of Proportion similar corresponding sides corresponding angles indirect measurement scale factor Finding Missing Measures in Similar Figures Find the value of x in each diagram. A. ABC JKL A 21 in. 14 in. J B 15 in. C K x in. L AB corresponds to. BC corresponds to. AC corresponds to. Complete the proportion. AB BC Complete the proportion where x represents KL. 21 x Set the cross products equal. Solve for x. 21x What is KL? B. DEGF PQSR D 02cm 0.2 F G E x cm Complete the proportion: DE PQ P R 6.4 cm S Q 1.6 cm x Complete the proportion where x represents GE. 0.2 x Set the cross products equal. Solve for x. 6.4x What is GE? x 37 Holt Algebra 1

17 2B Ready To Go On? Skills Intervention 2-8 Percents Find this vocabulary word in Lesson 2-8 and the Multilingual Glossary. Finding the Part Find 35% of 80. What operation can be substituted for the word of? If x represents the part, write an equation to represent the solution. x 35% 80 Write 35% as a number with a decimal point. 35% x 35% 80 Complete. x 80 Substitute the decimal value and operation into the equation and multiply. x 35% of 80 is. Vocabulary percent Finding the Percent What percent of 300 is 660? If 300 represents the whole, 660 represents the Complete the proportion: part whole percent 100 and x represents the percent. x 100 Set the cross products equal. Solve for x. 66,000 x 66,000 x % of 300 is 660. x Finding the Whole 21 is 60% of what number? What operation can be substituted for the word of? If x represents the whole, write an equation to represent the solution % Write 60% as a number with a decimal point. 60% x 21 x To isolate the variable, both sides by. x 21 x x 21 is 60% of. 38 Holt Algebra 1

18 2B Ready to Go On? Problem Solving Intervention 2-8 Percents Percents compare numbers to 100. The percent of a number can be found by writing and solving a proportion. There are 62 girls who participate in athletics. This represents 20% of the total number of girls in the school. How many girls are enrolled in the school? Understand the Problem 1. How many girls participate in athletics? 2. What percent of girls participate in athletics? Write 20% as a fraction. 20% Make a Plan 4. What do you need to determine? 5. Complete the proportion: part If g represents the number of girls, complete the proportion: g 20 Solve 7. Solve the equation by multiplying the products. 8. Solve the equation for g. 9. The number of girls enrolled in the g 20 20g 20g 20 g school is. Look Back 10. To check your answer, substitute your answer into the proportion from Exercise 6 and find the cross products Are the cross products equal? (62) (20) 39 Holt Algebra 1

19 2B Find these vocabulary words in Lesson 2-9 and the Multilingual Glossary. Vocabulary Ready To Go On? Skills Intervention 2-9 Applications of Percent commission principal interest tip sales tax Simple Interest Find the simple interest paid annually for 2 years on a $1200 loan at a 12% interest rate. The formula for simple interest is: Interest (Principal)(Rate)(Time) or I PRT. In this situation, $1200 is the principal, 2 years is the, and % is the rate. Substitute the numbers into the formula: I % Write the percent as a number with a decimal point. 12% Rewrite the formula using the decimal form of the rate: I 1200 What operation do you use to simplify the right side of the equation? How much money is paid in simple interest? I Estimating with Percents Estimate the tax on a $45,320 purchase when the tax rate is 11%. Round 45,320 to the nearest thousand. To make it easier to find 11% of a number, use the fact that 11% 10% %. 11% of 45,000 is the same as % of 45,000 1% of 45,000. To find 10% of 45,000 move the decimal To find 1% of 45,000 move the decimal point place to the left. point places to the left. 10% of 45,000 is. 1% of 45,000 is. 11% of 45,000 What is the estimated tax on the cost of the purchase? 40 Holt Algebra 1

20 2B Ready to Go On? Problem Solving Intervention 2-9 Applications of Percent There are many applications of percents. Among them are calculating commission, tax, tip, and interest. Laurie earns $6.50 per hour plus a 4% commission on her women s clothing sales. Find Laurie s total pay for a 40-hour week if her sales totaled $21,500. Understand the Problem 1. How much does Laurie earn per hour? 2. How many hours did Laurie work? 3. What percent commission does Laurie earn on top of her hourly wage? 4. How much were Laurie s sales total? Make a Plan 5. What do you need to determine? 6. Let T represent Laurie s total pay. Write an equation to represent the situation. Hourly Rate Number of hours Percent Commission Total Sales Total Pay T Solve 7. Find the amount of money earned from the hourly wage. 8. Write 4% as a number with a decimal point. 4% 9. Find the amount of money earned from commission. 10. Simplify the equation to solve for T ,500 T T 11. Laurie s total pay is. T Look Back 12. To check, subtract the answer in Exercise 7 from the answer in Exercise 11. Total Pay Money earned from hourly wages Money earned from commission 13. Is your answer the same as the money earned from commission? 41 Holt Algebra 1

21 2B Find these vocabulary words in Lesson 2-10 and the Multilingual Glossary. Vocabulary Ready To Go On? Skills Intervention 2-10 Percent Increase and Decrease percent change percent increase percent decrease discount markup Finding Percent Increase or Decrease Find each percent change. Tell whether it is a percent increase or decrease. A. from 72 to 90 Complete the formula for the percent of change: percent change original amount Is going from 72 to 90 an increase or a decrease? To find the amount of change, subtract: 90 What is the amount of change? Substitute the values into the formula: percent change Simplify the fraction. Write the fraction as a decimal. Write the decimal as a percent. amount of change original amount What is the percent increase? % B. from 40 to 26 Is going from 40 to 26 an increase or a decrease? To find the amount of change, subtract: 40 What is the amount of change? Substitute the values into the formula: percent change Simplify the fraction. Write the fraction as a decimal. Write the decimal as a percent. amount of change original amount What is the percent decrease? % What is the original amount? What is the original amount? 42 Holt Algebra 1

22 2B Ready to Go On? Problem Solving Intervention 2-10 Percent Increase and Decrease When an amount has been increased or decreased, the percent of change can be calculated. You can determine this percent change by dividing the amount of the change by the original amount. Carl purchased a pair of shoes for $68. This price was a 25% markup from the wholesale price. What was the wholesale price? Understand the Problem 1. How much did the shoes cost Carl? 2. What percent is the markup? Write this percent as a decimal. 3. Is the wholesale price more or less than the price that Carl paid? Make a Plan 4. What do you need to determine? 5. Let w represent the wholesale price of the shoes. Write an equation to represent this situation. Wholesale price of the shoes Wholesale price of the shoes Percent markup as a decimal Carl s Price w w Solve 6. Solve the equation by isolating x. 7. The wholesale price of the w w shoes was. w 68 w 68 w Look Back 8. Substitute the solution for w into the equation you wrote in Exercise 5. Wholesale price of the shoes Wholesale price of the shoes Percent markup as a decimal Carl s Price 9. Does your solution make the equation true? Holt Algebra 1

23 2B Ready to Go On? Quiz 2-6 Rates, Ratios, and Proportions 1. Last month, the ratio of DVD s to VHS tapes sold at a video store was 8:3. Sixty VHS tapes were sold. How many DVD s were sold? 2. Jorge cycled 48 miles in 4 hours. What is his cycling rate in miles per minute? Find the unit rate. 3. An 8-oz can of corn costs $ Brian can type 990 words in 30 minutes. Solve each proportion x 3 6. k r x Applications of Proportion Find the value of n in each diagram. 9. LMN QRP 10. UVWX GHJK M 12 cm N L 8 cm R 6 cm Q n cm P U V 5.4 yd X W 2.7 yd H G 1.8 yd K J n yd 44 Holt Algebra 1

24 2B 2-8 Percents Ready to Go On? Quiz continued 11. Find 25% of Find 160% of is what percent of 60? 14. What percent of 125 is 400? is 75% of what number? % of what number is 64? 17. On a class field trip, a parent-volunteer is responsible for 18 students. This represents 15% of the total number of students on the field trip. How many students are on the field trip? 2-9 Applications of Percent 18. Lauren earns $500 each month plus 3% commission on her car sales. Find Lauren s total salary for the month if her car sales total $115, Estimate a 15% tip on a $ restaurant bill Percent Increase and Decrease Find each percent change. Tell whether it is a percent increase or decrease. 20. from 50 to from 48 to from 300 to from 8.8 to Perla marks up her inventory 20% to sell to her customers. One of her customers purchased a bottle of hairspray for $ How much did Perla pay for the same bottle of hairspray? 45 Holt Algebra 1

25 2B Ready to Go On? Enrichment Compound Interest When investing money at a simple interest rate, the interest rate is applied only to the original principal amount in computing the amount of interest. To calculate simple interest, use the formula I PRT where I is the interest, P is the principal, R is the rate, and T is the number of years the principal has been invested. When investing money at a compounded interest rate, the interest rate is paid on the original principal and on the accumulated past interest. Compare Simple Interest with Compound Interest 1. Rosie invests $50 for 3 years at a 5% simple interest rate. Fill in the table. Principal Rate Time (years) Interest Interest Principal Rosie invests $50 for 3 years at a 5% compound interest rate. Fill in the table. Principal Rate Time (years) Interest Interest Principal The formula for compound interest is A P(1 r ) n, where P is the principal, r is the interest rate, n is the number of years the money is invested, and A is the amount of money accumulated from principal and interest over a period of time. 3. Which type of interest earns an investment more money? Answer each of the following questions. 4. Linda invests $1000 at a 8% compound interest rate for 6 years. How much money will she have in her account in 6 years? Round your answer to the nearest dollar. 5. George invests $12,000 at a 3% compound interest rate for 10 years. How much money will he have in his account in 10 years? Round your answer to the nearest dollar. 6. Lynn now has $ in her savings account. She invested her money at a 4% compound interest rate for 5 years. How much did she originally invest? Round your answer to the nearest hundred dollar. 46 Holt Algebra 1