Properties of Equality. Solving Equations

Transcription

1 Unit ~ Contents Algebra Beauty and Awe ~ China Order of Operations Using Parentheses to Translate From Word Phrases to Math Combining Like Terms Multiplying Terms Quiz Magic Squares Mean, Median, and Mode Dividing Terms The Distributive Property The Distributive Property and Mental Math Graphing Solutions Graphing Solutions Using a Vertical Number Line Quiz Durer s Beautiful Square Recognizing and Representing Values Algebraically Equations Solving Equations The Distributive Property and Division Mathematically Related Values Review for Test Test The Game of Nim

2 .1 Equations An equation is an algebraic sentence stating that two epressions are equal. The equation is perhaps the single most powerful tool in algebra, providing a way to find the values for variables. An equation works like a balance scale. Whatever operation is done to one side must also be done to the other to maintain the equality or balance Properties of Equality Using a, b, and c to represent real numbers, if a = b, then... a + c = b + c a c = b c a c = b c a c = b c...using the addition property of equality, the same number can be added to (or subtracted from) both sides of an equation....using the multiplication property of equality, both sides of an equation can be multiplied (or divided) by the same nonzero number. As long as the same value is added, subtracted, multiplied, or divided on both sides of an equation, the equality is maintained. Solving Equations Equations can be solved using the properties of equality. The foundational rule for solving equations is this: whatever operation is done to one side of the equation must be done to the other side. To Solve an Equation 1. Simplify each side of the equation.. Eliminate any constant from the variable side of the equation using addition property of equality.. Undo multiplication and division using the multiplication property of equality. 4. Check the solution by substituting it for the variable in the original equation..11 Recognizing and Representing Values Algebraically ~

3 Eample 1. Solve the equation + = = 17 + = 17 Original equation. subtracted from both sides to eliminate the constant from the variable side. = 14 Simplified. 14 = Both sides divided by to change the variable coefficient to 1. = 7 Simplified. Check: (7) + = 17 7 substituted for in the original equation 17 = 17 Simplified. Because this statement is true, the answer to the equation is correct. Eample. Solve the equation 5 64 = = ( 61) Original equation = ( 61) added to both sides to eliminate the constant from the variable side. 5 = Simplified. 5( ) = 5() Both sides multiplied by 5 to change the variable 5 coefficient to 1. = 15 Simplified. Check: = ( 61) 15 substituted for in the original equation. 61 = 61 Simplified. Because this statement is true, the answer to the equation is correct. 4 ~ Algebra I Unit

4 Eample. Solve the equation ( + 6) + 1 = 4. ( + 6) + 1 = 4 Original equation = = = 4 17 Multiplication of parentheses distributed. Like terms combined. 17 subtracted from both sides to eliminate the constant from the variable side. 5 = 5 Simplified. 5 5 = Both sides divided by 5 to change the variable 5 5 coefficient to 1. = 5 Simplified. Check: (5 + 6) + (5) 1 = 4 5 substituted for in the original equation. 4 = 4 Simplified. Because this statement is true, the answer to the equation is correct. Solve = 1. 4 = = = = = ( 1) = (5 ) + 7 = ( ) = 7 Review Assign variable(s) to the unknown(s). Then translate the epression the average of three different numbers 11. the product of four-thirds π and the radius cubed 1. five more than double the height Translate into symbols. Do not simplify or solve.. 1. a number equals five less than the product of seven and another number 14. the difference of the absolute value of negative eight and four equals a number 15. five times the sum of si and four equals a number.1 Equations ~ 5

5 Simplify a Write the word(s) for each definition. 4. a number factor in front of a variable (1.4) 5. the number under the radical sign for which we are finding the root (1.14) 6. a group of numbers, variables, and symbols of mathematical operations (1.4) 7. To subtract a number, add its. (1.7) 8. the number above the radical sign that indicates the root called for (1.14) Write the property the equations illustrate = 0 0. (9 + 11) + 7 = 9 + (11 + 7) ( 5) = 0 Write the answer.. Hikers leave the rim of the Grand Canyon and begin the 8,000-foot descent to the canyon floor. 45 feet into the canyon, they stop for a rest. How high are they above the canyon floor? (1.7). Does (6 )(6 4) = 6( 4)? (1.4) 4. Evaluate both (6 )(6 4) and 6( 4). (1.4) Evaluate the epressions if =, y = 4, and z = yz z + y 7. yz 5 Combine like terms in the epressions y yz + 6yz yz y y Solve = = ( + 4) = The great buildings constructed during the Qin Dynasty, around 00 B.C., show an advanced understanding of the use of area, volume, and proportion in their engineering. 6 ~ Algebra I Unit

6 .1 The Distributive Property and Division Just as the distributive property is used for multiplication, it can also be used for division. Division is usually written in fractional form and the divisor (denominator) distributed to each term in the numerator. Each term in the numerator must be divided by the denominator. Eample 1. ( + 6) and Original problem set up as a fraction Division distributed to each term in the numerator Factors canceled Simplified Eample. (15y y 4 + 1) y 15y y y Original problem written as a fraction. 15y y 515 y 1 y y 4 1 y + Division distributed to each term of the numerator. y y y + 1 Factors canceled. y y Simplified. y Problems do not need to be rewritten in fractional form. However, it is necessary to divide each term in the dividend by the divisor. Simplify (5 10) 5. (1 + 8) 4. (4 18y + 7) 4. (z z) z 5. (4k + 6k) k 6. ( ) 4 7. (4 8) 8. (6p + p) p 9. (d 1) 4d.1 The Distributive Property and Division ~ 7

7 Review Solve = = ( ) + 9 = 54 Simplify yz(7yz + z ) 14. 5(y y + ) y 8 Evaluate {5[5 + (4 7)] } ( 5) Rewrite with positive eponents s y Duplicate these number lines on graph paper and graph the given solutions y = y = = = Find the mean, median, and mode Ingmar read advertisements for houses to rent at these monthly rates; $65, $1,00, $85, $995, $750, $850, $ ~ Algebra I Unit

8 Multiply the terms z y zy 6. y z 7. 0yz yz Divide the terms y 54yz 9. yz 4 y z 0. 97yz 97yz Combine the numbers ( 9) ( 7) + ( 9) Write the property the equations illustrate = = 1 Assign variable(s) to the unknown(s). Then translate the epression one-third the product of the base and height 7. double the product of the length and width Write the answer. 8. The mission team determined that 1 cow + goats + 0 chickens would be a good gift package to help poor families recovering from a flood. If the mission distributed 1 of these gift packages, how many of each animal did they distribute? Evaluate 1(1 cow + goats + 0 chickens) to calculate the answer. (.8) 9. According to one respected time line, Adam was born in 4004 B.C. and he died when he was 90 years old. Lamech (Noah s father) was born in 10 B.C. Was Adam still alive when Lamech was born? When Lamech was 0 years old? (1.7) Simplify ( 7 6 y 1 4 y ) 7 y 41. ( y + 6y ) y 4. (16 y + 18y + 0) 4y.1 The Distributive Property and Division 9

9 .14 Mathematically Related Values Some problems have more than one unknown value, but the values are mathematically related to each other. When values are mathematically related, they can be mathematically epressed using just one variable and the relationship. Eample 1. Tina picked three more quarts of blueberries than her younger sister Joan. Two values are epressed here the number of quarts Tina picked, and the number of quarts Joan picked. The phrase three more than shows the mathematical relationship between the two values. If Joan s quarts are represented with, then Tina s quarts can be represented with +. Joan s Quarts ( j) Tina s Quarts ( j + ) Eample. Travis caught four times as many fish as Edward. Again, two values are epressed here the number of fish Travis caught, and the number of fish Edward caught. This time the phrase four times as many shows the mathematical relationship between the two values. So if Edward s fish is represented with, then Travis fish can be represented with 4. Edward s Fish (e) Travis Fish (4e) Eample. A yardstick (6 in) is divided into two unequal parts. Two values are epressed here the lengths of the two parts. The word divided suggests that the parts are related by division. In this situation the values are actually related by subtraction, because if the first piece is represented with (), the second part is what remains after is taken away from 6, or (6 ). First Part () Second Part (6 ) 40 ~ Algebra I Unit

10 Eample 4. Three consecutive integers. Three values are epressed here three integers in a row (such as 1, 1, 14). Anywhere on the number line each integer is one more than the one before it. This describes the relationship between the values. If the first integer is represented with, the net integer would be one more, or + 1. The third integer would be another one more, or +. First Integer Second Integer Third Integer () ( + 1) ( + ) There is no rule that says which unknown is used for the starting variable. What is important is showing the mathematical relationships between the values correctly. In the last eample, could have been used to represent the third integer. Then 1 would represent the second integer and the first. Both ways show the correct relationship between consecutive integers. Label and algebraically represent the values Angela is five times the age of her sister Julie. Let represent Julie s age.. John is one-third the age of his father.. Michael used more gallons of gas than Elliot. 4. The cost of bicycle A is $17 less than bicycle B. Let represent the cost of bicycle B. 5. A meter stick (100 cm) is divided into two unequal parts. 6. Craig had twice as many pennies as nickels. 7. Four consecutive integers. Review Distribute the division and simplify (8 4) 9. (16m 8n ) 4m n 10. (6 5y 14) y Simplify ( 8 4y) 1. 7( + ) 0 10 Divide the terms y 5 z 0y z yz 5 7 yz 15. 1r 6 s 4 rs Combine like terms in the epressions yz + 5yz 0 yz yz y 6y + 4y + y.14 Mathematically Related Values 41

11 Translate into symbols. Do not simplify or solve a number equals seven more than si divided by another number 19. the quotient of forty-eight and eight, diminished by the square of a number 0. the square of two, times the difference of ten and five, equals a number plus two Write the property the equations illustrate = = Multiply or divide. 1.9; ( 9) Write the answer. 6. Woodsmen have long known that you can estimate the temperature of the air by counting the chirps number of times a cricket chirps in one minute and using the formula F = Estimate the temperature if a cricket chirps 11 times in one minute. (1.4) 7. The sum of two numbers is negative. The first number is positive. What must be true about the second number? (1.6) Simplify. Leave in eponential form (6 5 ) Assign variable(s) to the unknown(s). Then translate the epression three consecutive even integers (such as 14, 16, 18). volume equals the length times the width times the height Solve..1. = = 5. 7 = 6 Evaluate ( ) 1 7. ( 7) Find the mean, median, and mode. Round to the nearest tenth On Monday, Tuesday and Wednesday, Janessa gathered 16 eggs from her chickens each day. On Thursday and Friday, she collected 17 eggs each day. On Saturday she gathered 15 eggs. Label and algebraically represent the values A -mile trail is divided into two unequal sections. 4. There are one-half as many girls as boys in the ninth grade class. 4 ~ Algebra I Unit