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UNIT 5. EXPRESSIONS USING ALGEBRA 5. 1 ALGEBRAIC EXPRESSIONS Objective Learn how to replace numbers with letters called variables. Summary Algebra is an area of mathematics where letters are used to represent numbers. You can use algebra to solve mathematical problems. A variable is a letter used to represent an unknown number, for example, x. In algebra, letters are used when numbers are not known, unknown. r + 2s means an unknown number 'r', plus 2 lots of an unknown number 's'. Q1. Say that 'g' is the cost of child admission, and 'k' is the cost of adult admission to the zoo. a) How much does it cost for the Khan family of 3 children and 3 adults to visit the zoo? b) Write an algebraic expression for the cost for the Norman family of 5 children and 4 adults to visit the zoo. http://kent.skoool.co.uk/content/keystage3/maths/pc/learningsteps/VRBLC/launch.html (1) 1. Write variable expressions to represent word problems: 1. Annie has m buttons. Trey has 36 more buttons than Annie. How many buttons Trey has? 2. Katrina has w hair bows. Katrina's sister has 7 more hair bows than Katrina. How many hair bows Katrina's sister has? 3. Hannah had t books. Then she went to a book bought 89 more books. How many books Hannah has now? 4. Jeremy has z trading cards. Martha has 37 more trading cards than Jeremy. How many trading cards Martha has? 5. There were 5 frogs in a pond. k of the frogs hopped away. How many frogs are in the pond now? 1 6. Colin has m small spoons and 25 big spoons. Write an expression that shows how many spoons Colin has. 7. Emilie has w trading cards. Tori has 61 more trading cards than Emilie. Write an expression that shows how many trading cards Tori has. 8. Mason planted 94 fewer trees than Larry. Larry planted k trees. Write an expression that shows how many trees Mason planted. 9. Meredith has 40 buttons. She gives away p buttons. Write an expression that shows the number of buttons Meredith has left. shows how many apples Wyatt picked. 12. Alex had 6 CDs until he won r more in a contest. How many CDs Alex has now? 13. The advert says that the cost of a party is 50€ plus 5€ per person, a) So what would be the cost of a party for 10? b) So what would be the cost of a party for 20? c) How do you work out the cost if you don't know the number of people who will attend the party? 14. Alexa is 12 years old. How old will she be in n years time? 10. Jake ate u out of 83 gumdrops. Write an expression that shows how many gumdrops Jake has left. 15. Roland weighs 70 kilograms, and Mark weighs k kilograms. Write an expression for their combined weight. 11. Nicole picked 97 apples. Wyatt picked b fewer apples than Nicole. Write an expression that 16. A car travels down the freeway at 55 km/h. Write an expression for the distance the car will have travelled after h hours. http://kent.skoool.co.uk/keystage3.aspx?id=65 > Multiplication 27 > Scene 1 – 4 (2) Algebraic terms, like 2s or 8y, leave the multiplication signs out. So rather than '2 · s', write 2s, rather than '8 ·× y' write 8y. A string of numbers and letters joined together by mathematical operations such as + and - is called an algebraic expression 2. Write using algebraic expressions: a) The double of a number plus 5 units. d) The cube of a number. b) The triple of a number less 6 units. e) A number plus its square. c) The square of a number. f) Half of a number minus 3. 2 Solving problems in algebra depends on your ability to represent missing or unknown quantities. Representing unknown quantities is easy to do if you know the "language". For example, what operation is meant by the phrase more than ? If you guessed addition, then you are right! This skill of "translating" between words and mathematical operations just takes a little vocabulary drill. Below are some of the most common phrases used in problem solving, together with the operations they represent. Study these relationships and you should be a whiz at algebraic representation. ADD add sum more than increased by exceeds in all total plus SUBTRACT subtract difference *less than decreased by diminished by minus fewer reduced by MULTIPLY multiplied by of product times double twice triple DIVIDE divide quotient divided equally per ratio of * be careful using "less than" - it reverses the order of things Also, be careful of the placement of commas in statements. In the statement "the sum of a and b, divided by 3" the comma indicates that the answer is (a+b)/3 and not a + b/3. EXAMPLES 1. two more than a number 2+x 2. five less than three times a number 3x - 5 notice how this changed the order 3. seven times a number, increased by 4 4. six decreased by 5 times a number 5. 7x + 4 6 - 5x Given 2x - 4, write a verbal expression that matches this mathematical expression. Some possible verbal answers: twice a number decreased by 4 double a number minus 3 Numerical value of an algebraic expression 1. What is the value of 3(2x - 5) when x = 4? a) 6x – 5 b) 9 c) 30 2. What is the value of 4(y + 10) when y= 5? a) y + 40 b) 40 c) 60 3. What is the value of 6g + 10k when g = 5 and k = 2? a) 31 b) 40 c) 50 4. What is the value of 2x2- y when: a) x=0, y =1 b)x=-1, y=-2 5. What is the value of x2 + 1 when x=2? 6. Calculate the following expressions: a + bc2 − c(b + ac)2 = a=2 b=3 c = (-3) 3n2 + 4n = n=1 b − (ac)2 + (3 − b) = a=4 b = (-2) c=3 n(n+3) = n = -2 n2 (5+ 4n) = n3 + n2 + n = n = -1 n=2 4 5.2 Monomials In algebra, a letter can be used to represent a number that you do not know. This gives us algebraic terms, like 2x and 5y, called monomials. Monomial is a product of two or some factors, each of them is either a number, or a letter, or a power of a letter. For example, 3 a 2 b 4 , b d 3 , – 17 a b c are monomials. A single number or a single letter may be also considered as a monomial. The numerical factor is called a coefficient. Degree of monomial is a sum of exponents of the powers of all its letters. Q1. Complete: Monomial -8xy2 3a2b3 Coefficient Letters 4 -9 2 3 x3y2 abc Degree b2 When algebraic terms are put together by mathematical operations such as + and - we get an algebraic expression, like 2x + 5y +3z. We can simplify algebraic expressions by collecting 'like terms'. Like terms Monomials are called similar or like terms if they are identical or differed only by coefficients. -25a2 and 8a2 are like terms. 3z2 and -4z are not like terms Monomials that have the same letter are called like terms (monomis semblants). Only like terms can be added (+) or subtracted (-). Examples: Terms 7x Why are they "Like Terms" x -2x (1/3)xy2 -2xy2 6xy2 because the variables are all x because the variables are all xy2 5 Q2. Write a like term in each monomial from Q1. To simplify an expression, we collect like terms. Like terms include letters and numbers that are the same. Look at the expression 4x + 5x -2 - 2x + 7 To simplify: The x terms can be collected together to give 7x. The numbers can be collected together to give 5. So 4x + 5x -2 - 2x + 7 simplified is 7x + 5. Remember that terms are separated by + and - signs, and these are always attached to the front of a term. Example So 9b, -7b and 13b are like terms, but 6t, 5x and -11z are not like terms. When like terms are added and subtracted it is called simplifying. Different terms To answer some exam questions you will have to simplify an expression that has many different terms or letters. Have a look at this typical exam question. You will notice that there are three different terms in this question: x, y and z. Q3. Simplify the expressions 5x + 3y - 6x + 4y + 3z = 5a + 3b - 3a - 5c + 4b = a) b) c) d) e) f) 2 x 2 3x 2 8 x 3 14 x 3 6x9 4x9 12 x 5 4 x 5 11x 4 7 x 4 13x 6 8 x 7 2 3 1 3 x x 3 2 5 h) x 2 x5 1 i) y 2 2 y 2 y 2 2 1 g) 3 y 7 y 7 2 y 7 3 g) 6 5.3 Operating with monomials 5.3.1 Adding and subtracting like terms The Khan family and the Norman family visit the zoo together - there are 8 children and 7 adults in the group. Because there are more than 10 people, the families can take advantage of a special offer (1 child can be admitted free of charge). As before let's use g for the cost of child admission, and k for the cost of adult admission. Cost for the Khan family = 3g + 3k Cost for the Norman family = 5g + 4k Offer = - g Total cost = 3g + 3k + 5g + 4k - g = 7g + 7k http://kent.skoool.co.uk/keystage3.aspx?id=65#24_30 > 25. Terms (3) 1. Simplify these expressions: 1) a+a+a = 2) 2b+3b = 3) a+b+a+b+c = 4) 6c+2d+3c+3d + and – signs A term, like a number, belongs with the sign that sits in front of it. So in the expression 2k - g, g belongs with the - sign that sits in front of it, so it is - g; and 2k belongs with the + sign that sits in front of it, making it + 2k. Remember: Collect all the like terms together, eg, re-write the expression 3g + 2k + 5g + 4k - g with all the g s and all the k s together: 3g + 5g - g + 2k + 4k When you add or subtract terms, keep each term with their + or - sign. In algebra, we often get very long expressions that we need to make simpler. Simpler expressions are easier to solve. http://kent.skoool.co.uk/keystage3.aspx?id=65 > 26. Expressions (4) 7 Questions 1. Simplify these expressions: a) x + 5 + 3x- 7 + 9x+ 3 - 4x = b) 4x + y - 2x + 6y = c) 3g + 2k + 5g + 4k - g = 2. When the expression 3a + 9b - 2b is simplified, it becomes: a) 12ab - 2b b) 3a + 7b c) 3a + 11b 3. When the expression 14m - 6k - 5m + 3k + 4k is simplified, it becomes: a) 9m + k b) 9m + 13k c) 19m - 7k Remember: Like terms with powers can be added or subtracted but only if the powers are the same. x is not the same as x2, so they cannot be added together. Example Rearrange the expression so that like terms are next to each other Simplify: 3x + 5 + x2 - 2x + 2x2 - 1 = x2 + 2x2 + 3x - 2x + 5 – 1 = 3x2 + x + 4 Simplify People usually write the expression with the 'x2' term first, then the 'x' term and finally the number term. In general, in decreasing order. Questions 4. Simplify: a) b) c) d) e) f) g) 5 + 4x2 - 3 + 2x + x2 - 4x = 5x + 2x = −3y2 + 4y2 = 2ab2 − a2b = 5x2 + 7x = 5x3 −6x + 7x − x3 −x + 4x3= −x2 + x + x2 + x3 + x = 8 h) i) j) k) 2x3 −(x3 −3x3) = 8x2 −x + 9x + x2 = 8xy2 −5x2y + x2y −xy2= −3x + 7y −(8y + y −6x) = 5. Write true or false: a) a + a = 2a b) 2a + a = 2 a2 c) 2a − a =2 d) 2a −2 = a e) 2a −b = 2 · (a −b) f) 2a + 3a = 5a g) 2a + 3b = 5ab h) 2a2 = 4a 6. Complete : a) (6x2 −4x + 7) + (8x2 +2x-3) = b) (5x3 + 3x2 −10) − (6x2 −4x + 7) = c) (9x3 + x2 − 6x + 4) + (x2 −4x + 3) - (6x2 −2x + 7) = 5.3.2 Multiplying and dividing monomials Multiplying letters Monomials can be multiplied in the same way as numbers. a · a = a2, b · b = b2 etc. Remember that 2a is not the same as a2 2a = 2 · a a2 = a · a In general, am · an = a(m + n) Questions Simplify: a) a2 · a3 = b) p4 · p2 = Remember that x is the same as x1 when using this rule. eg, n3 · n = n(3 + 1) = n4 9 Multiplying numbers and letters Multiply letters and the numbers separately: 2 · 3a means 2 · 3 · a which is 6a 4a · 5a means 4 · a · 5 · a which is 4 · 5 · a · a = 20a2 Question 1. Simplify b) 2x2· 4x3 ·5x6= a)2p · 3p2 = c) −3x ·(−2x)· 7x 4 d) 7x 3 ·5x · 9x4= 2.Complete: · 3x 4x 5x 2 2x 6x 2 2x 2 Objective: Understand how to multiply variables and terms in an expression. Summary When multiplying terms, remember the following steps: 1. First, multiply the numbers. 2. Then multiply the variables. 3. Write the numeral first, followed by the variable. Dividing letters Algebraic expressions can be divided in the same way as numbers. x8 x5 3 x x4 x3 x In general, a m : a n a m n 10 Dividing numbers and letters Divide letters and the numbers separately: 6x5 6 x5 · 2 3x 3 2 2 x 2x 6 12 x 12 x 6 4 x 3 · 5 15 x 3 15 x 3 2 3 2 3 7x y 7 x y 7 xy 3 · 4x 4 x 4 http://kent.skoool.co.uk/keystage3.aspx?id=65#24_30 > Algebraic fractions (5) Simplify: a) 15x3 : 5x2 = b) −8x3y2 : 2x2y = c) 10x4 yz2 : 5xyz = d) −9a : 3a e) −3y2 : 4y2= f) −10x3y2 : x2y = g) 2ab2 :a2b= h) 10x3 : 2xy2= i) 2x2 · x3 · 3x5 : (−6x)= j) 8x4 : (2x2 + 2x2) = k) (5y 3 −2y 3) : (3xy2) = l)15x3 : 5x2= m) −8x3 y2 : 2x2y= n) 10x4yz 2 : 5xyz= 5.4 Polynomials A monomial or the sum of two or more monomials is called a polynomial. monomial that makes up a polynomial is referred to as a term of the polynomial. 6x5 - 3x4 - x3 - 9x + 7 Each CONSTANT TERM TERMS 11 A term is a number, variable or the product of a number and variable(s). Examples of terms are : A coefficient is the numeric factor of your term. Here are the coefficients of the terms listed above: Term Coefficient 3 5 2 A constant term is a term that contains only a number. In other words, there is no variable in a constant term. There are special names for polynomials with 1, 2 or 3 terms: Degree of polynomial is the most of degrees of monomials, forming this polynomial. Give the degree of the following polynomial: 2x5 – 5x3 – 10x + 9 This polynomial has four terms, including a fifth-degree term, a third-degree term, a first-degree term, and a constant term. This is a fifth-degree polynomial. Give the degree of the following polynomial: 7x4 + 6x2 + x This polynomial has three terms, including a fourth-degree term, a seconddegree term, and a first-degree term. There is no constant term. This is a fourth-degree polynomial Q1. Give the degree of the following polynomial: 6x5 - 3x4 - x3 - 9x + 7 2x3 - 3x2 - 1 12 Evaluation "Evaluating" a polynomial is the same as evaluating anything else: you plug in the given value of x, and figure out what y is supposed to be. For instance: Evaluate P(x)= 2x3 – x2 – 4x + 2 at x = –3 I need to plug in "–3" for the "x", remembering to be careful with my parentheses and the negatives: P(-3) = 2(–3)3 – (–3)2 – 4(–3) + 2 = 2(–27) – (9) + 12 + 2 = –54 – 9 + 14 = –63 + 14 = -49 Always remember - to be careful with the minus signs! If we see a variable standing alone (it has no coefficient, no number next to it) then we assume that there is an invisible one (1) standing there: x2 = 1x2 Q2. Simplify like terms in these polynomials, order them in decreasing order and give the degree in each polynomial. a) P (x ) = 5x3 −x + 7x3 −x2 + 8x −2 b) Q (x ) = 12 + x2 + 7x −x4 −8 + 3x2 c) R (x ) = 9x −4x2 −6 −10x + 1 d) S (x ) = 4x2 −x3 + 4x3 −x5 + 8 −x2 Q3. Evaluate Q(x ) = 3x4 −2x3 + x2 −5, at x= -1 Q4. Find the value of a to get P (x ) = ax2 −3x + 5verifies P (2) = 3. Q5. If P (x ) = 3x4 −2x3+ x2 −5, calculate: a) P (1) + P(0) −P (−2) b) 2 ·P (2) + 3·(P (−1)) http://www.youtube.com/watch?v=IDpnNnjFB1c Adding and Subtracting Polynomials (6) Q6. Given : A (x) = 2x3 −3x2 + x −7 B (x) = x3 + 7x2 − 4x C (x) = −2x2 + x − 5 Then calculate: a) A (x) + B (x) + C (x) c) A (x) − B (x) b) B (x) + C (x) d) A (x) −B (x) −C (x) 13 Multiplying polynomials MONOMIAL · POLYNOMIAL Tools for Dealing with Distributive Property a(b + c) = ab + ac Rules for Exponents Multiplying Signed Numbers (+) • (+) = (+) (+) • (-) = (-) (-) • (+) = (-) (-) • (-) = (+) Monomials Example 1: monomial • monomial (4x3) • (3x2) = (4 • 3) • (x3 • x2) = 12 • x5 = 12x5 Notice that the factors were regrouped and then multiplied. Also, multiply powers with the same base by adding the exponents. = x2 + 4x This problem requires the distributive property. You need to multiply each term in the parentheses by the monomial (distribute the x across the parentheses). Example 2: monomial • binomial Notice the distributive property at work again. Example 3: monomial • trinomial = 2x3 + 6x2 + 8x Again, the distributive property is needed along with the rule for multiplying powers. Example 4: monomial • polynomial = 3x5 - 9x4 + 18x3 - 15x2 Multiply –3x·(4x2 – x + 10) = –3x(4x2) – 3x(–x) – 3x(10) = –12x3 + 3x2 – 30x x3(x4 + 5a) = x3(x4 + 5a) = x7 + 5ax3 2 x3 5x5 2 x 4 4 x3 6 x 2 x 7 14 POLYNOMIAL· POLYNOMIAL Multiply (x + 3)(x + 2) 1. "Distributive" Method: The most universal method. Applies to all polynomial multiplications, not just to binomials. Start with the first term in the first binomial - the circled blue X. Multiply (distribute) this term times EACH of the terms in the second binomial. Now, take the second term in the first binomial - the circled red +3 (notice we take the sign also). Multiply this term times EACH of the terms in the second binomial. Add the results: x•x + x•2 + 3•x +3•2 x² + 2x +3x + 6 = x² + 5x + 6 (x + 5)(a − 6) = x(a − 6) + 5(a − 6) = ax − 6x + 5a − 30 (x − 3)2 = (x − 3)(x − 3) = x(x − 3) − 3(x − 3) = x2 − 3x − 3x + 9 = x2 − 6x + 9 (2x + 3)(x2 − x − 5) = (2x)(x2 − x − 5) + (3)(x2 − x − 5) = = (2x3 − 2x2 − 10x) + (3x2 − 3x − 15) = 2x3 + x2 − 13x − 15 We take the 2 terms of the first bracket and multiply both of them by the second bracket. 15 (x + 3)(x² + 2x + 4) = x³ + 2x² + 4x + 3x² + 6x + 12 = x³ + 2x² + 3x² + 4x + 6x + 12 Group like terms. = x³ + 5x² + 10x + 12 Q6. Multiply: a) (3x + 4) ·2= Combine like terms. c) (4x 2 + x −2) · (−5) = b) (x −2) · 4x = d) (x2+ 3x −6) · (−3x3) = Q7. Multiply and simplify: a) (x + 3) · (x −2) = b) (2x −6) · (3x + 5) = c) (4 −6x + 3x2) · (−2 −x + x2 ) = d) (x −5)2 = e) (4 + a)2 = f) (2x + 3y)2 = g) x 5 x · x 2 x h) x 4 2 x 3 1 · x 3 5 i) (3x3 5x 2 3x 1)(3x3 5x) j) (5x5 3x 4)(7 x3 2 x 1) k) (25x7 5x 2)( x 4 3x 2 3) l) ( x3 2 x 2 2)(3x 4 5 x3 4 x 2 4) m) x 1 ( x 2) 2 n) ( x 3) 2 x 2 3x 1 2 Solutions: 9 x 6 15 x5 24 x 4 22 x3 15 x 2 5 x 8 6 5 4 3 2 j) 35 x 10 x 5 x 21x 28 x 6 x 5 x 4 11 9 7 5 4 3 2 k) 25 x 75 x 75 x 5 x 2 x 15 x 6 x 15 x 6 i) 16 3x 7 x 6 14 x5 14 x 4 14 x3 16 x 2 8 3 2 m) x 4 x 5 x 2 5 4 3 2 n) 4 x 24 x 41x 9 x 17 x 3 l) Q8. Roll the two cubes, record the polynomials on the answer sheet, then MULTIPLY the polynomials. Q9. Practice with Multiplying Polynomials http://www.regentsprep.org/Regents/math/ALGEBRA/AV3/PracPoly.htm Special Pattern Binomials The following are special multiplications involving binomials that you will want to try to remember. Be sure to notice the patterns in each situation. You will be seeing these patterns in numerous problems. Don't panic! If you cannot remember these patterns, you can arrive at your answer by simply multiplying with the distributive method. These patterns are, however, very popular. If you can remember the patterns, you can save yourself some work. Squaring a Binomial - multiplying times itself (a + b)² = a² + 2ab + b² (a - b)² = a² - 2ab + b² Notice the middle terms in both of these problems. In each problem, the middle term is twice the multiplication of the terms used to create the binomial expression. Example 1: Example 2: (x + 3)² = (x + 3)(x + 3) = x² + 3x + 3x + 9 Distributive method = x² + 6x + 9 Notice the middle term. (x - 4)² = (x - 4)(x - 4) = x² - 4x - 4x + 16 Distributive method = x² - 8x + 16 Again, notice the middle term. 17 Product of Sum and Difference (notice that the binomials differ only by the sign between the terms) (a + b)(a - b) = a² - b² Notice that there appears to be no "middle" term to forma a trinomial answer, as was seen in the problems above. When multiplication occurs, the values that would form the middle term of a trinomial actually add to zero. Example 3: (x + 3)(x - 3) = x² - 3x + 3x - 9 Distributive method = x² - 9 *Notice how the middle term is zero. Example 4: (2x + 3y)(2x - 3y) = 4x² - 6xy + 6xy - 9y² Distributive method = 4x² - 9y² * Again, notice how the middle term is zero. Q10. Calculate using patterns showed above: a) (a + 4)2 = b) (2x + 3)2 = c) (b + 12)2 = d) (3x + 5y)2 = e) (x + 8y2)2 = f) (5ab + 3b2)2 = g) (50 – x)2 = h) (ab – 20)2 = i) (5x – 11)2 = j) ( 1 3 42 a –a ) = 2 k) (2xy – 7x)2 = 18 l) (3x – 4y3)2 = m) (3 + x) · (3 – x) = n) (x4 – 3y) · (x4 + 3y) = o) (4 + 3x) · (3x – 4)= p) (6ab + 3b2) · (6ab – 3b2) = q) (– 2x – 10) · (10 – 2x) = 5.5 Formulas What is an Equation? An equation says that two things are equal. It will have an equals sign "=" like this: x + 2 = 6 That equations says: what is on the left (x + 2) is equal to what is on the right (6) So an equation is like a statement "this equals that" What is a Formula? A formula is a special type of equation that shows the relationship between different variables. http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/formulaeact.shtml (7) We often use formulas without even noticing. For example, we might convert miles to km by multiplying by 1.6, or find the circumference of a circle by multiplying pi by the diameter. km = 1.6 x miles is an example of a formula. L = · d (circumference of a circle) 19 Question. A rectangle has a width of x and a length of 2x. Write down a formula for the perimeter (P) in terms of x. P= TEST. Using formulas: 1. What's the formula connecting speed, distance and time? a) s = d·t b) s = d/t c) s = d + t 2. If F = ma, what's the value of F when m = 5 and a = 3? a) 8 b) 15 c) 53 3. If A = 3b - c2, find the value of A when b = 4 and c = 2 a) 30 b) 16 c) 8 4. If a = b - c, then: a) c = b – a b) c = a + b c) c = a - b b) c) t(v - u) 5. If v = u + at, then a =: a) 5.6 Extracting common factors http://kent.skoool.co.uk/keystage4.aspx?id=317 > Factors 1 (8) In Maths we multiply into brackets as follows: 3x 5 x 2 2 x 4 15x 3 6 x 2 12 x We can also carry out the reverse process (extracting common factors). An expression can be simplified by extracting out the common factor. 20 When factoring polynomials, first look for the largest monomial which is a factor of each term of the polynomial. Factor out (divide each term by) this largest monomial. 15 x 3 6 x 2 12 x = 5·3·x·x·x - 2·3·x·x + 4·3·x = = 5·3·x·x·x - 2·3·x x + 4·3·x =3x · 5x 2 2 x 4 1. Complete: a) 5x 10 y 5x ...... c) a 2 b 6a 2 b 2 a 2 b...... ...... b) 6 x 2 8x 2 x3x ...... d) 2ab 5ac a 2 a2b ..... ..... 2. Extract common factor: a) 3abc 5bc 7abcd 2bcd b) 5x 2 y 3 z 7 3x 8 y 4 z 11x 5 y 3 z 4 c) 14 x 3 y 2 z 4 7 x 5 y 3 z 7 21x 2 y 5 z 3 d) 54n 4 m 6 18n 3 m 5 27n 7 m 3 e) 5 x 10 xy 5 x 2 y f) 3x 5 y 4 9 x 2 y 3 3xy 3 y g) 5 y 2 x 15 yx 2 y 3 x 4 h) 6 x 2 y 2 9 x 3 y 6 27 xy3 i) 9 x 3 y 12 x 2 y 2 18xy3 j) 4x + 8y = k) 15x2y3 + 10xy2 = ACTIVITIES 1. Do the questions carefully. (39 questions) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 21 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 2. Vocabulary Here are some words used in algebra: brackets unknown equation value expression variable formula solution term Complete the following statements to show that you understand the meaning of these words. 1. An unknown is _______________________________________________________. 2. A variable is _________________________________________________________. 3. The difference between an equation and an expression is ______________________ ______________________________________________________________________. 4. A formula is _________________________________________________________. 5. In algebra, brackets are _______________________________________________ . 6. Another word for solution is ____________________________________________ . 7. You find the value of the formula when you ________________________________ 8. A term is a ___________________________________________________________ 22 3. Expressions This is an expression: We can write this in words as: n+3 Add three to a number. Now write the following expressions in words. 1. 5 − n 2. n + 1 3. 2n − 3 4. 6 − 2n n 5. 3 6. n2 C. Inverse What’s the meaning of inverse? Operation 1. addition (+) 2. division (:) 3. multiplication (·) 6. square (x2) 7. square root ( ) 8. subtraction (-) The words on the right are all operations used in algebra. Write the inverse operation alongside each term. The first one has been done for you. 3. Evaluate 6a + 4b + 5c when: 1) a=1, b=1, c=1 2) a=2, b=-1, c=3 4. Multiply out the brackets and simplify: a) 3(a+4) = c) 5(x-2)+4(y-2) = Inverse operation _______________ _______________ _______________ _______________ _______________ _______________ 3) a=3, b=-3, c=-4 b) 4(a+2)+3(b+1) = d) 3(2x-3y+z)+2(3x+5y)= 5. Word problems as equations. When converting word problems to equations, certain "key" words tell you what kind of operations to use: addition, multiplication, subtraction, and division. The table below shows some common phrases and the operation to use. Word Operation Example sum Addition difference subtraction product times less than multiplication multiplication subtraction total Addition more than Addition The sum of my age and 10 equals 27. The difference between my age and my younger sister's age, who is 11 years old, is 5 years. The product of my age and 14 is 168. Three times my age is 60. Seven less than my age equals 32. The total of my pocket change and 20 dollars is $22.43. Eleven more than my age equals 43. As an equation 23 24