Math 2 Variable Manipulation Part 2 Student 1 COMPARING EQUATIONS (EQUAL TO) When comparing two problems, set part of the equations to be the same, then set the rest of the equation to be equal and solve. Look at the problem to determine similarities. In many problems, set the base to be the same. Example: If 9 2x-1 = 3 3x+3, then x -? Solution: Express the left side of the equation so that both sides have the same base: Now that the bases are the same, just set the exponents equal: 4x 2 = 3x + 3 4x 3 = 3 + 2 X = 5 Other problems will have similar numerators or denominators. Make them exactly the same and set the rest of the equation to be equal. Example: What is the sum of all solutions to 4x = 4x x 1 2x+2? Solution: Since the numerators are both 4x, set the denominators equal to each other x 1 = 2x + 2 -x 2 -x - 2-3 = x 1. In real numbers, what is the solution of the equation 8 2x + 1 = 4 1 - x? 2. Which real number satisfies (2 n )(8) = 16 3? 3. If 3 8x = 81 3x - 2, what is the value of x? 4. If a, b, and c or consecutive positive integers and 2 a x 2 b x 2 c = 512, then a + b + c =? 5. If 4 9 y 11 = 4 9 11, then y =?
ADDING AND SUBTRACTING POLYNOMIALS To add or subtract polynomials, combine like terms: (3x 2 + 5x - 7) - (x 2 + 12) = (3x 2 - x 2 ) + 5x + (-7-12) = 2x 2 + 5x - 19 6. (6a 12) (4a + 4) =? 2 7. x 2 + 60x + 54 59x 82x 2 is equivalent to: 8. For all x, x 2 (3x 2) + 2x(4x 1) =? 9. What polynomial must be added to x 2 2x + 6 so that the sum is 3x 2 + 7x? MULTIPLYING MONOMIALS To multiply monomials, multiply the numbers and the variables separately: 2a x 3a = (2 x 3)(a x a) = 6a 2 10. Which of the following is an equivalent form of x + x(x + x + x)? a. 5x b. x 2 + 3x c. 3x 2 + x d. 5x 2 e. x 3 + x 11. Which of the following is NOT a solution of (x 5)(x 3)(x + 3)(x + 9) = 0? a. 5 b. 3 c. -3 d. -5 e. -9 SOLVING "IN TERMS OF" To solve an equation for one variable in terms of another means to isolate the one variable that you are solving for on one side of the equation, leaving an expression containing the other variable on the other side. To solve 3x - 1Oy = -5x + 6y for x in terms of y, isolate x: 3x - 10y = -5x + 6y 3x + 5x = 6y + 1Oy 8x = 16y x= 2y
12. For all pairs of real numbers M and N where M = 6N + 5, N =? 3 13. If the expression x 3 + 2hx - 2 is equal to 6 when x = -2, what is the value of h? 14. What is the value of the expression (x y) 2 when x = 5 and y = 1? 15. How many ordered pairs (x,y) of real numbers will satisfy the equation 5x 7y = 13? Equivalent Forms Besides plugging in numbers to get an answer, solving in terms of x or another variable, there are ways to simplify equations and express them as equivalent equations. This can be accomplished in many ways. Equations will be equivalent as long as all mathematical rules are followed. Example: Which expression below is equivalent to w(x (y + z))? a. wx wy wz b. wx wy + wz c. wx y + z d. wx y - z e. wxy + wxz Solution: To find an equivalent for the given expression, use the distributive property. First, evaluate the inner parentheses according to the order of operations, or PEMDAS. Distribute the negative sign to (y + z) to get w(x y z). Next, distribute the variable w to all terms in parentheses to get wx wy wz or A. Choice B fails to distribute the negative sign to the z term. Choices C and D only distribute the w to the first term. Choice E incorrectly distributes wx to the (y + z) term. 16. Which of the following is a simplified form of 4x 4y + 3x? a. x(7 4y) b. x y + 3x c. -8xy + 3x d. 7x 4y e. -4y -x 17. 2r + 4s is equivalent to: 3 5 18. When y = x 2, which of the following expressions is equivalent to y?
4 19. Which of the following is (are) equivalent to the mathematical operation a(b c) for all real numbers a, b, and c? I. ca ba II. ab ac III. (b c)a a. II only b. I and II only c. I and III only d. II and III only e. I, II and III 20. Which of the following is always equal to y(3 y) + 5(y 7)? a. 8y -35 b. 8y -7 c. y 2 + 8y 7 d. y 2 + 8y 35 e. 8y 3 35 21. If W = XYZ, then which of the following is an expression for Z in terms of W, X, and Y? a. XY W W b. XY c. WXY d. W XY e. W + XY 22. The expression a[(b c) + d] is equivalent to: a. ab + ac + ad b. ab ac + d c. ab ac + ad d. ab c + d e. a c + d UNDERSTANDING CHANGES IN VALUES Another way to evaluate algebraic equations is to look at what happens to the equation as you change the values of the variables. For example, if you increase the value of the denominator, a number will decrease in value. One way to look at these types of questions is to plug in various numbers and see what happens to the answer. Example: Let n equal 3a + 2b -7. What happens to the value of n if the value of a increases by 2 and the value of b decreases by 1? a. It is unchanged. b. It decreases by 1. c. It increases by 4. d. It decreases by 4. e. It decreases by 2. Solution: Set a = 1 and b = 1 and 3a + 2b -7 becomes 3(1) + 2(1) 7 = 3 + 2 7 = -2. Then increase a by 2 and decrease b by 1 and which will mean to set a = 3 and b = 0 and the equation becomes 9 + 0 7 = 2. The answer increased by 4. Try a second set of numbers to verify the same result. Set a = 2 and b = 2 to get 3(2) + 2(2) -7 = 6 + 4 7 = 3. Increase a by 2 and decrease b by 1 which becomes a = 4 and b = 1. Substituting into the equation to get 12 + 2 7 = 7. This answer is also increased by 4. So the answer is c.
5 23. Let x = 2y + 3z -5. What happens to the value of the x if the value of y decreases by 1 and the value of z increases by 2? a. It decreases by 2 b. It is unchanged c. It increases by 1 d. It increases by 2 e. It increases by 4 24. In the equation r = 4 (2+k), k represents a positive integer. As k gets larger without bound, the value of r: a. Gets closer and closer to 4 b. Gets closer and closer to 2 c. Gets closer and closer to 0 d. Remains constant e. Gets larger and larger 25. IF ghjk = 24 and ghkl = 0, which of the following must be true? a. g > 0 b. h > 0 c. j = 0 d. k = 0 e. l = 0 LOGIC Some problems do not require a formula or an equation, they just take some thinking to figure out what the problem is asking and how to solve it. 26. Fifty high school students were polled to see if they owned a cell phone and an MP3 player. A total of 35 of the students own a cell phone, and a total of 18 of the students own an MP3 player. What is the minimum number of student who own both a cell phone and an MP3 player? 27. Assume that the statements in the below are true. All students who attend Tarrytown High School have a student ID Amelia does not attend Tarrytown High School. Carrie has a student ID. Tracie has a student ID. Joseph attends Grayson High school. Michael is a high school student who attends Tarrytown High School. Considering only these statements, which of the following statements much be true? a. Michal has a student ID. b. Amelia is not a high school student. c. Carrie attends Tarrytown High School. d. Traci attends Tarrytown High School. e. Joseph does not have a student ID.
6 28. If m, n, and p are positive integers such that m + n is even and the value of (m + n)2 + n + p is odd, which of the following must be true? a. m is odd b. n is even c. p is odd d. If n is even, p is odd e. If p is odd, n is odd 29. Which of the following is true for all consecutive integers m and n such that m < n? a. m is odd b. n is odd c. n m is even d. n 2 m 2 is odd e. m 2 + n 2 is even MULTIPLYING BINOMIALS-FOIL To multiply binomials, use FOIL: First, Outer, Inner, Last: Example: (x + 3)(x + 4) First multiply the First terms: x x x = x 2 Next the Outer terms: x x 4 = 4x. Then the Inner terms: 3 x x = 3x. And finally the Last terms: 3 x 4 = 12. Then add and combine like terms: x 2 + 4x + 3x + 12 = x 2 + 7x + 12. 30. The expression (x + 4)(x 2) is equivalent to: 31. The expression (3x 4y 2 )(3x + 4y 2 ) is equivalent to: 32. The expression (4z + 3)(z 2) is equivalent to:
7 FACTORING OUT A COMMON DIVISOR A factor common to all terms of a polynomial can be factored out. All three terms in the polynomial 3x 3 + 12x 2-6x contain a factor of 3x. Pulling out the common factor yields 3x(x 2 + 4x - 2). Remember that if you factor a term out completely, you are still left with 1: in the expression 6x 2 + 9x + 3, you can factor a 3 out of everything. You're left with 3(2x 2 + 3x + 1). 33. x 2 x + px 34. 36x 2-64y 4 35. 3x 3 + 27x 2 + 9x FACTORING THE DIFFERENCE OF SQUARES One of the test maker's favorite classic quadratics is the difference of squares. a 2 - b 2 = (a - b)(a + b) Example: x 2-9 factors to (x - 3)(x + 3). FACTORING THE SQUARE OF A BINOMIAL There are two other classic quadratics that occur regularly on the ACT: a 2 + 2ab + b 2 = (a + b) 2 a 2-2ab + b 2 = (a - b) 2 For example, 4x 2 + 12x + 9 factors to (2x + 3) 2 and n 2-10n, + 25 factors to (n - 5) 2. Recognizing a classic quadratic can save a lot of time on Test Day-be on the lookout for these patterns. (HINT: Any time you have a quadratic and one of the numbers is a perfect square, you should check for one of the patterns.) Square binomials can also be treated the same way as regular foil problems after writing out the binomial. Example: (2x 4) 2 (2x -4)(2x -4) = 4x 2-8x - 8x + 16 = 4x 2 16x + 16 This answer is in the form a 2-2ab + b 2 = (a - b) 2 36. For all n, (3n+ 5) 2 =? 37. For all x,(3x + 7) 2 =? 38. ( 1 3 a b)2 =?
FACTORING OTHER POLYNOMIALS-FOIL IN REVERSE To factor a quadratic expression, think about what binomials you could use FOIL on to get that quadratic expression. To factor x 2-5x + 6, think about what First terms will produce x what Last terms will produce + 6, 2 and what Outer and Inner terms will produce -5x. If there is no number in front of the first term, you are looking for two numbers that add up to the middle term and multiply to the third term. So here, you'd want two numbers that add up to -5 and multiply to 6. (Pay attention to sign-negative vs. positive makes a big difference here!) The correct factors are (x - 2)(x - 3). You can also solve for x for each factor. In this case x = 2 and x = 3. Note: Not all quadratic expressions can be factored. These expressions will be covered in Math 2. 8 39. What is the product of the two real solutions of the equation 2x = 3 - x 2? 40. Which of the following is a factored form of the expression 5x 2 13x 6? a. (x 3)(5x + 2) b. (x 2)(5x 3) c. (x 2)(5x + 3) d. (x + 2)(5x 3) e. (x + 3)(5x 2) 41. If x 2 45b 2 = 4xb, what are the 2 solutions for x in terms of b? 42. What values of x are solutions for x 2-2x = 8? 43. What is the product of the 2 solutions of the equation x 2 + 4x 21 = 0? 44. Which of the following is a polynomial factor of x 2 2x 24? a. x 4 b. x + 4 c. x + 6 d. 6 - x e. x 45. What is the smallest value of x that satisfies the equation x(x + 4) = -3?
9 SIMPLIFYING AN ALGEBRAIC FRACTION Simplifying an algebraic fraction is a lot like simplifying a numerical fraction. The general idea is to find factors common to the numerator and denominator and cancel them. Thus, simplifying an algebraic fraction begins with factoring, which often involves reverse-foil. To simplify x2 x 12, first factor the numerator and denominator: x 2 x 12 x 2 9 x 2 9 = (x 4)(x+3) (x 3)(x+3) Canceling x + 3 from the numerator and denominator leaves you with (x 4) (x 3). 46. If x is any number other than 3 and 6, then (x 3)(x 6) (3 x)(x 6) =? 47. For x 2 169, (x 13)2 x 2 169 =? 48. For all a 0 and b 0, a+b b(a+b) 2a(a+b) =? 49. If A + B = 7A+2B and A, B, and x are integers greater than 1, than what must x equal? 30 105 x 50. For all x 1, x2 2x+1 x 1 is equal to?
10 ADDING ALGEBRAIC FRACTIONS (GREATEST COMMON FACTOR IN DENOMINATOR) To add fractions, the denominators must be the same. Therefore, as a common denominator choose the LCM of the original denominators. The least common denominator must be a multiple of the denominator of each of the fractions. First find the least common denominator. Then, convert each fraction to an equivalent fraction with the same denominator by multiplying each fraction by the missing factors. Finally, add the fractions together and reduce as necessary. Example: Add 3 + 4 + 5 ab bc cd Solution: The least common denominator is abcd because each of these variables are included in at least one of the denominators. To change 3 into an equivalent fraction with denominator abcd, simply multiply ab by ab the factors it is missing, namely cd. Therefore, we must also multiply 3 by cd. That accounts for the first term in the numerator. To change 4 into an equivalent fraction with denominator abcd, multiply bc by the factors bc it is missing, namely ad. Therefore, we must also multiply 4 by ad. That accounts for the second term in the numerator. To change 5 into an equivalent fraction with denominator abcd, multiply cd by the factors it is cd missing, namely ab. Therefore, we must also multiply 5 by ab. That accounts for the last term in the numerator. Remember that EACT factor of the original denominators must be a factor of the common denominator. 3 + 4 + 5 ab bc cd = 3cd+4ad+5ab abcd Example: When adding fractions, a useful first step is to find the least common denominator (LCD) of the fractions. What is the LCD for these fractions? 2, 13, 2 3 2 x5 5 2 x7x11 3x11 3 a. 3 x 5 x 7 x 11 b. 3 2 x 5 2 x 7 x 11 c. 3 2 x 5 2 x 11 3 d. 3 2 x 5 2 x 7 x 11 3 e. 3 2 x 5 3 x 7 x 11 4 Solution: The least common denominator must be a multiple of the denominator of each of the three given fractions. Take the factors of each denominator (3, 5, 7, 11) to the highest powers they appear: 3 2 (first fraction), 5 2 (second fraction), 7 (second fraction), and 11 3 (third fraction). This results in 3 2 x 5 2 x 7 x 11 3, choice D. Choice E is the product of all three denominators and its wrong because choice D is smaller and still a multiple of all three denominators; in other words, both are common denominators but D is the least. 51. What is the least common denominator for the expression below? 1 + 1 + 1 a 2 x b x c b 2 x c b x c 2 52. What is the solution for the expression below? 4y 6(x 2)(x+5) + 2y 3x(x+5) 53. What is the solution for the expression below? 3-2 4(x 2 1) (x 1)(x 2)
11 SOLVING A QUADRATIC EQUATION To solve a quadratic equation, put it in the form of ax 2 +bx + c = 0 - in other words, set it equal to 0. Then factor the left side (if you can), and set each factor equal to 0 separately to get the two solutions. To solve x 2 + 12 = 7x, first subtract 7x from both sides of the equation, which gives you x 2-7x + 12 = 0. Then use reverse- FOIL to factor the left side: (x - 3)(x - 4) = 0 x - 3 = 0 or x - 4 = 0 x = 3 or 4 Sometimes the left side might not be obviously factorable. You can always use the quadratic formula. Just plug in the coefficients a, b, and c from ax 2 + bx+ c = 0 into the formula: b ± b 2 4ac 2a To solve x 2 + 4x + 2 = 0, plug a = 1, b = 4, and c = 2 into the formula: 4 ± 4 2 4 1 2 2 1 4 ± 8 2 = -2 ± 2 Whether you use reverse-foil or the quadratic equation, you will almost always get two solutions, or roots, to the equation. 54. For a certain quadratic equation, ax 2 + bx + c = 0, the solutions are x = 0.75 and x = - 0.4. Which of the following could be factors of ax 2 + bx + c? a. (4x - 3) AND (5x + 2) b. (4x - 2) AND (5x + 3) c. (4x + 2) AND (5x - 3) d. (4x + 3) AND (5x - 2) e. (4x + 3) AND (5x + 2) 55. For what nonzero whole number k does the quadratic equation y 2 + 2ky + 4k = 0 have exactly one real solution for y? a. 8 b. 4 c. 2 d. -4 e. -8 56. In the equation x 2 + mx + n = 0, m and n are integers. The only possible value for x is 3. What is the value of m? 57. The table below gives values of the quadratic function f for selected values of x. Which of the following defines the quadratic function f? x 0 1 2 3 F(x) -6-5 -2 3 a. f(x) = x 2 6 b. f(x) = x 2 + 6 c. f(x) = 2x 2-10 d. f(x) = 2x 2-6 e. f(x) = 2x 2 7
58. Let S be the set of all integers that can be written as 2n 2 6n, where n is a nonzero integer. Which of the following integers is in S? a. 6 b. 30 c. 46 d. 64 e. 80 12 SOLVING A SYSTEM OF EQUATIONS You can solve for two variables only if you have two distinct equations. If you have one variable, you only need one equation, but if you have two variables, you need two distinct equations. Two forms of the same equation will not be adequate. If you have three variables, you need three distinct equations, and so on. Combine the equations in such a way that one of the variables cancels out. To solve the two equations 4x + 3y = 8 and x + y = 3, multiply both sides of the second equation by -3 to get: - 3x - 3y = -9. Now add the equations; the 3y and the -3y cancel out, leaving x = -1: 4x + 3y = 8 +(-3x - 3y = -9) X = -1 Plug that back into either one of the original equations and you'll find that y = 4. 59. If mn = k and k =x 2 n, and nk 0, which of the following is equal to m? a. 1 b. 1/x c. x d. x e. x 2 60. What is the largest possible product for 2 odd integers whose sum is equal to 32? 61. When x/y = 4, x 2 12y 2? 62. The larger of two numbers exceeds twice the smaller number by 9. The sum of twice the larger and 5 times the smaller number is 74. If a is the smaller number, which equation below determines the correct value of a? a. 5(2a + 9) + 2a = 74 b. 5(2a - 9) + 2a = 74 c. (4a + 9) + 5a = 74 d. 2(2a + 9) + 5a = 74 e. 2(2a - 9) + 5a = 74
63. The average of a and b is 6 and the average of a, b, and c is 11. What is the value of c? 13 64. The sum of the real numbers a and b is 13. Their difference is 5. What is the value of ab? 65. If x + y = 13 and 2y = 16, what is the value of x? 66. If x + 4y = 5 and 5x + 6y = 7, then 3x + 5y =? 67. The costs of carriage rides of different lengths, given in half miles, are shown in the table below: Number of 5 6 7 10 half miles Cost $8.00 $8.50 $9.00 $10.50 Each cost consists of a fixed charge and a charge per half mile. What is the fixed charge? 68. Marcia makes and sells handcrafted picture frames in 2 sizes: small and large. It takes her 2 hours to make a small frame and 3 hours to make a large frame. The shaded triangular region shown below is the graph of a system of inequalities representing weekly constraints Marcia has in making the frames. For making and selling s small frames and l large frames. For making and selling s small frames and l large frames, Marcia makes a profit of 30s + 70l dollars. Marcia sells all the frames she makes. For every hour that Marcia spends making frames in the second week of December each year, she donates $3 from that week s profit to a local charity. This year, Marcia made 4 large frames and 2 small frames in that week. Which of the following is closest to the percent of that week s profit Marcia donated to the charity? 69. For what value of n would the following system of equations have an infinite number of solutions? 3a + b = 12 2a + 4b = 3n a. 4 b. 9 c. 16 d. 36 e. 48
70. A system of linear equations is shown below. 4y 2x = 8 4y + 2x = 8 Which of the following describes the graph of the system of linear equations in the standard (x,y) coordinate plane? a. A single line with positive slope b. A single line with negative slope c. Two distinct intersecting lines d. Two parallel lines with positive slope e. Two parallel lines with negative slope 14 71. The equations below are linear equations of a system where a, b, and c are positive integers. ay + bx = c ay bx = c Which of the following describes the graph of at least 1 such system of equations in the standard (x,y) coordinate plane? I. 2 parallel lines II. 2 intersecting lines III. A single line a. I only b. II only c. III only d. I or II only e. I, II, or III 72. If x = 6a + 3 and y = 9 + a, which of the following expresses y in terms of x? a. y = x + 51 b. y = 7x + 12 c. y = 9 + x d. y = 57+x 6 e. y = 51+x 6 73. Which of the following equations expresses z in terms of x for all real numbers x, y, and z, such that x 5 y and y 3 = z? a. z = x b. z = 3/5 x c. z = 3x 5 d. z = x 8 e. z = x 15 74. If (a + b) 2 = 25 and (a - b) 2 = 45, then a 2 + b 2 =? 75. Two small pitchers and one large pitcher can hold 8 cups of water. One large pitcher minus one small pitcher constitutes 2 cups of water. How many cups of water can each pitcher hold?
15 76. Margie is responsible for buying a week's supply of food and medication for the dogs and cats at a local shelter. The food and medication for each dog costs twice as much as those supplies for a cat. She needs to feed 164 cats and 24 dogs. Her budget is $4240. How much can Margie spend on each dog for food and medication? FUNCTIONS SUBSTITUTING INTO ANOTHER FUNCTION 77. Given f(x) = 3x + 5 and g(x) = x 2 x + 7, which of the following is an expression for f(g(x))? a. 3x 2-3x + 26 b. 3x 2-3x + 12 c. x 2 - x + 12 d. 9x 2 + 25x + 27 e. 3x 2 + 21 78. If f(x) = 2x 2 + 3, then f(x + h) =? 79. For the 2 functions f(x) and g(x), tables of values are shown below. What is the value of g(f(-1))? 80. Let a function of 2 variables be defined by f(a,b) = ab (a b). What is the value of f(8,9)? 81. If the function f satisfies the equation f(x + y) = f(x) + f(y) for every pair of real numbers x and y, what is (are) the possible value(s) of f(1)? a. Any real number b. Any positive real number c. 0 and 1 only d. 0 only e. 1 only 82. Consider the functions f(x) = x and g(x) = 7x + b. In the standard (x,y) coordinate plane, y = f(g(x)) passes through (4,6). What is the value of b? 83. Given f (x) = 4x + 1 and g(x) = x 2 2, which of the following is an expression for f g(x)? a. x 2 + 4x + 1 b. x 2 + 4x 1 c. 4x 2 7 d. 4x 2 1 e. 16x 2 + 8x 1
Answer Key 1. -1/8 2. 9 3. 2 4. 9 5. 11 6. 2(a 8) 7. -81x 2 + x + 54 8. 9x 2 5x +2 9. 2x 2 + 9x - 6 10. 3x 2 + x 11. -5 12. M 5 6 13. -4 14. 36 15. Infinitely many 16. 7x 4y 17. (10r+12s) 15 18. x 2 19. D 20. D 21. B 22. C 23. E 24. C 25. E 26. 3 27. A 28. D 29. D 30. x 2 + 2x - 8 31. 9x 2 16y 4 32. 4z 2 5z 6 33. -x(x + 1 p) 34. 4(9x 2-16y 4 ) 35. 3x(x 2 + 9x + 3) 36. 9n 2 + 30n + 25 37. 9x 2 + 42x + 49 38. 1 9 a2 2 ab + b2 3 39. -3 40. (x 3)(5x + 2) 41. 9b or -5b 42. -2 and 4 43. -21 44. x + 4 45. -3 46. -1 47. (x 13) (x+13) 1 48. b 2a 49. 210 50. x 1 51. a 2 x b 2 x c 2 52. 8yx 4y 6x(x 2)(x+5) 53. 3(x 2) 8(x+1) 4(x+1)(x 1)(x 2) 54. (4x - 3) AND (5x + 2) 55. 4 56. 9 57. f(x) = x 2 6 58. 80 59. x 2 60. 255 61. 4y 2 62. 2(2a + 9) + 5a = 74 63. 21 64. 36 65. 5 66. 6 67. $5.50 68. 14% 69. 16 70. C 71. B 72. E 73. E 74. 35 75. 2 76. $40 77. A 78. 2x 2 + 4xh + 2h 2 + 3 79. 3 80. 73 81. Any real number 82. 8 83. C 16