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Study Island Algebra 1 Critical Topics Simplify Expressions 1. Simplify. A. 10x2 - 26xy - 12y2 B. 10x2 + 11xy - 6y2 C. 10x2 - 15xy - 6y2 D. 10x2 + 26xy + 12y2 Patterns 2. What is the explicit formula for this sequence? 45 , 48 , 51 , 54 , 57 , ... A. an = 48 - 3n B. an = 42 + 3n C. an = 47 - 2n D. an = 43 + 2n Patterns 3. What is the explicit formula for this sequence? -4 , 4 , 12 , 20 , 28 , ... A. an = -12 + 8n B. an = -11 + 7n C. an = 4 - 8n D. an = 3 - 7n Quadratic Equations 4. Solve for x in the following equation. A. B. C. D. Linear Equations 5. (1/5) (2x + 7) = 4 What is the first step when solving the above equation for x? A. Multiply each side by 5. B. Subtract 2x from each side. C. Divide each side by 5. D. Subtract 5 from each side. Patterns 6. Define the sequence below recursively. an = 5n - 2 A. an = (an-1)5 with a1 = 3 B. an = an-1 - 5 with a1 = 3 C. an = 8/3·an-1 with a1 = 3 D. an = an-1 + 5 with a1 = 3 Patterns 7. What is the recursive formula for this sequence? 33 , 26 , 12 , -9 , -37 , ... A. a1 = 33, an = 7(an-1) - (n-1) B. a1 = 33, an = an-1 - 7(n) C. a1 = 33, an = an-1 - 7(n-1) D. a1 = 33, an = an-1 - 7 Simplify Expressions 8. Simplify. A. B. C. D. Linear Equations 9. Solve for x. A. x = 9 B. x = 8 C. x = 7 D. x = 6 Linear Equations 10. Solve for x. A. x = -4 B. x = 5 C. x = -6 D. x = -5 Systems of Equations & Inequalities 11. The following system of equations is graphed below. -2x - 2y = 0 -5x + 2y = 0 Find the solution to the system. A. x = 0, y = 1 B. x = 1, y = 0 C. x = -5, y = 0 D. x = 0, y = 0 Simplify Expressions 12. Simplify the following expression. 3a3 - 4ab + 3a3 A. 6a6 - 4ab B. 6a3 - 4ab C. 9a3 - 4ab D. 2a3 - 4b Systems of Equations & Inequalities 13. Solve for x in the two equations below using substitution. A. B. C. D. Linear Equations 14. Which algebraic expression is equivalent to the expression below? 8(4x + 4) + 2x A. 34x + 32 B. 34x + 4 C. 6x + 32 D. 32x + 32 Patterns 15. What is the recursive formula for this sequence? 42 , 38 , 30 , 18 , 2 , ... A. a1 = 42, an = an-1 - 4(n-1) B. a1 = 42, an = an-1 - 4 C. a1 = 42, an = an-1 - 4(n) D. a1 = 42, an = 4(an-1) - (n-1) Systems of Equations & Inequalities 16. The side of a square is 8 inches longer than the side of an equilateral triangle. The perimeter of the square is 48 inches more than the perimeter of the triangle. What are the lengths of a side of the square, s, and a side of the triangle, t? A. s = 8 inches; t = 16 inches B. s = 24 inches; t = 32 inches C. s = 40 inches; t = 32 inches D. s = 24 inches; t = 16 inches Quadratic Equations 17. Which algebraic expression is equivalent to the expression below? (x - 4)2 - (x + 4)2 A. -16x B. 2x2 + 32 C. 16x + 20 D. -2x2 - 16x - 32 Simplify Expressions 18. Simplify. (4x - 2y)2 + 16xy A. 16x2 + 4y2 B. 16x2 - 16xy + 4y2 C. 16x2 + 16xy + 4y2 D. 16x2 - 4y2 Quadratic Equations 19. Solve for x in the following equation. x2 + 3x = 28 A. x = 4; -8 B. x = -4; 7 C. x = 4; -7 D. x = -1; 28 Systems of Equations & Inequalities 20. The following system of equations is graphed below. 2x - 3y = 13 5x + 5y = -5 Find the solution to the system. A. x = 2, y = 3 B. x = 2, y = -3 C. x = -2, y = 3 D. x = -2, y = -3 Systems of Equations & Inequalities 21. Solve for y in the two equations below using substitution. A. B. C. D. Simplify Expressions 22. Simplify the following expression. 3(5y + 3z) - 9z A. 15y - 12z B. 8y C. 8y - 18z D. 15y Linear Equations 23. Solve for x. A. x = 78 B. x = -15 C. x = 52 D. x = 65 Quadratic Equations 24. Solve for x in the following equation. A. B. C. D. Systems of Equations & Inequalities 25. Find the solution to the system of equations given below using elimination by addition. 4x + 16y = 80 x + 2y = 10 A. B. C. D. x = 0, y = 5 x = -30, y = 20 x = 20, y = -5 x = 50, y = -20 Systems of Equations & Inequalities 26. Which set of inequalities best represents the graph shown above? y > -1/2x + 5 x > -2 A. y > -2 y > -1/2x + 5 x > -2 B. y > -2 C. y < -1/2x + 5 x > -2 y > -2 y < -1/2x + 5 x > -2 D. y > -2 Linear Equations 27. Solve for x. A. x = 3 B. x = 2 C. x = -2 D. x = 4 Simplify Expressions 28. (2x4y2z)(6x3y2z) A. 8x12y4z B. 12x12y4z C. 12x7y4z2 D. 8x7y4z2 Quadratic Equations 29. Which algebraic expression is equivalent to the expression below? -3(-x + 4)2 A. 3x2 - 24x + 48 B. -3x2 + 24x - 48 C. -3x2 - 48 D. x2 - 8x + 16 Simplify Expressions 30. Simplify. A. B. C. D. Patterns 31. Find the fifth term in the recursively defined sequence below if a1 = 1. an = 2·an-1 + 3 A. 29 B. 13 C. 125 D. 61 Quadratic Equations 32. Solve for x in the following equation. x2 + 10x + 24 = 0 A. x = -1; -24 B. x = -4; -6 C. x = 4; 6 D. x = 4; -6 Patterns 33. The first five terms of a sequence are given below. 34 , 36 , 38 , 40 , 42 , ... Determine which of the following formulas gives the nth term of this sequence. A. 32 + 2n B. 33 + n C. 35 - n D. 36 - 2n Simplify Expressions 34. Simplify. 5(4a - 5b) + 2(2a - 2b) A. 24a - 29b B. 24a - 21b C. 42a - 49b D. 24a - 7b Patterns 35. What is the explicit formula for this sequence? 11 , 0 , -11 , -22 , -33 , ... A. an = 22 - 11n B. an = -11 + 22n C. an = -11 - 22n D. an = 22 - 10n Linear Equations 36. Which value for x makes the sentence true? 9x + 4 = 22 A. x = 13 B. x = 4 C. x = 2 D. x = 9 Linear Equations 37. Solve for x. A. x = -7 B. x = 8 C. x = -9 D. x = -8 Quadratic Equations 38. Which algebraic expression is equivalent to the expression below? [-2x - 3 - (3x - 6)]2 A. x2 + 6x + 9 B. 25x2 + 90x + 81 C. x2 - 18x + 81 D. 25x2 - 30x + 9 Systems of Equations & Inequalities 39. Find the solution to the system of equations given below using elimination by addition. A. B. C. D. Quadratic Equations 40. Solve for x in the following equation. x(x - 10) = 11 A. x = 1; 11 B. x = -1; -11 C. x = 1; -11 D. x = -1; 11 Answers 1. A 2. B 3. A 4. B 5. A 6. D 7. C 8. B 9. B 10. D 11. D 12. B 13. B 14. A 15. A 16. D 17. A 18. A 19. C 20. B 21. A 22. D 23. D 24. A 25. A 26. D 27. B 28. C 29. B 30. B 31. D 32. B 33. A 34. A 35. A 36. C 37. D 38. D 39. A 40. D Explanations 1. 2. Each number in the sequence is 3 more than the previous number (3n). Use the first term to develop the nth term. 45 = ___ + 3(1) 45 = 42 + 3 an = 42 + 3n 3. Each number in the sequence is 8 more than the previous number (8n). Use the first term to develop the nth term. -4 = ___ + 8(1) -4 = -12 + 8 an = -12 + 8n 4. Factor the equation and solve for x. 5. Given (1/5) (2x + 7) = 4, we are trying to solve for, or isolate, x the simplest way possible. We would start by either distributing the 1/5 or multiplying by 5. Staying away from fractions is usually the simplest way. So, to get rid of the 1/5, multiply each side by 5. 6. Find the first four or five numbers in the sequence to identify a pattern. a1 = 5(1) - 2 = 3 a2 = 5(2) - 2 = 8 a3 = 5(3) - 2 = 13 a4 = 5(4) - 2 = 18 a5 = 5(5) - 2 = 23 Every term is 5 more than the previous term. So, an = an-1 + 5 would be the recursive definition of the given sequence. 7. When finding a recursive formula, examine how each term differs from the previous term. This sequence decreases by multiples of 7: first by 7, then 14, then 21, then 28,... Each term (an) is the value of the previous term (an-1) decreased by 7 times the number of the previous term (n-1). For instance, 26 = 33 - (7 × 1). In other words, current term = previous term - (7 × previous term number). Now, write it as a recursive formula. a1 = 33, an = an-1 - 7(n-1) 8. When an exponential expression is raised to another exponent, multiply the exponents. When dividing exponential expressions with the same base, subtract the exponents. Any number raised to the zero power is equal to 1. 9. 10. 11. The point of intersection of the two graphs gives the solution to the system of equations. The two lines intersect at (0, 0). So, x = 0 and y = 0 is the solution. 12. Simplify the expression by combining like terms. 3a3 - 4ab + 3a3 = 6a3 - 4ab 13. Solve the first equation for y. Now, substitute this answer for y into the second equation. Then, solve for x. 14. Use properties of arithmetic to transform the given expression. 8(4x + 4) + 2x 32x + 32 + 2x Distribute the 8. Combine like terms. 34x + 32 15. When finding a recursive formula, examine how each term differs from the previous term. This sequence decreases by multiples of 4: first by 4, then 8, then 12, then 16,... Each term (an) is the value of the previous term (an-1) decreased by 4 times the number of the previous term (n-1). For instance, 38 = 42 - (4 × 1). In other words, current term = previous term - (4 × previous term number). Now, write it as a recursive formula. a1 = 42, an = an-1 - 4(n-1) 16. The side of the square is 8 inches longer than the side of the equilateral triangle. s-t=8 The perimeter of the square is 48 inches more than the perimeter of the triangle. 4s - 3t = 48 To find the side lengths of the square and triangle, solve the system of equations below. s-t=8 4s - 3t = 48 In this case, solve by substitution. s-t =8 s=t+8 So, 4s - 3t = 48 4(t + 8) - 3t = 48 4t + 32 - 3t = 48 t = 16 and s=t+8 s = 16 + 8 s = 24. Therefore, the length of a side of the square is 24 inches, and the length of a side of the triangle is 16 inches. 17. Use properties of arithmetic to transform the given expression into one of the answer choices. (x - 4)2 - (x + 4)2 (x2 - 8x + 16) - (x2 + 8x + 16) -16x 18. 19. Factor the equation and solve for x. x2 + 3x = 28 x2 + 3x - 28 = 0 (x - 4)(x + 7) = 0 x = 4; -7 20. The point of intersection of the two graphs gives the solution to the system of equations. The two lines intersect at (2, -3). So, x = 2 and y = -3 is the solution. 21. Solve the first equation for x. Now, substitute this answer for x into the second equation. Then, solve for y. 22. Simplify the expression by using the distributive property and combining like terms. 3(5y + 3z) - 9z = 15y + 9z - 9z = 15y 23. Keep in mind that the goal is to get x by itself. 24. Factor the equation and solve for x. 25. Use elimination by addition to solve the system of equations. 4x + 16y = 80 x + 2y = 10 Start by eliminating the x term. To do this, multiply the second equation by -4, and then add the two equations together. 4x + 16y = 80 -4x - 8y = -40 8y = 40 y= 5 Next, substitute y = 5 into the original second equation, and solve for x. x + 2y = 10 x + 2(5) = 10 x + 10 = 10 x = 10 - 10 x=0 Therefore, the solution to the system of equations is x = 0, y = 5. 26. First look at the slanted line. It follows the path of y = -1/2x + 5. Notice that the shaded region is below the line and the line is solid. Therefore, the correct inequality for this part of the graph is y < -1/2x + 5. Next look at the vertical line. It follows the path of x = -2. Notice that the shaded region is to the right of the line and the line is solid. Therefore, the correct inequality for this part of the graph is x > -2. Finally, look at the horizontal line. It follows the path of y = -2. Notice that the shaded region is above the line and the line is dotted. Therefore, the correct inequality for this part of the graph is y > -2. 27. 28. When multiplying exponential expressions with the same base, add the exponents. So, break up the given terms into terms with common bases. (2x4y2z)(6x3y2z) = (2)(6)(x4)(x3)(y2)(y2)(z)(z) = 12x(4+3)y(2+2)z(1+1) = 12x7y4z2 29. Use properties of arithmetic to transform the given expression into one of the answer choices. -3(-x + 4)2 -3(x2 - 8x + 16) -3x2 + 24x - 48 30. When an exponential expression is raised to another exponent, multiply the exponents on each term. 31. a1 = 1 a2 = 2·a1 + 3 = 2·1 + 3 = 2 + 3 = 5 a3 = 2·a2 + 3 = 2·5 + 3 = 10 + 3 = 13 a4 = 2·a3 + 3 = 2·13 + 3 = 26 + 3 = 29 a5 = 2·a4 + 3 = 2·29 + 3 = 58 + 3 = 61 So, a5 = 61. 32. Factor the equation and solve for x. x2 + 10x + 24 = 0 (x + 4)(x + 6) = 0 x = -4; -6 33. A generic arithmetic sequence is of the following form, a , a + d , a + 2d , a + 3d , ... , a + (n - 1)d , ... where a is the first term, d is the common difference, and a + (n - 1)d is the nth term. In this case, the first term is 34 and the common difference is 2. Therefore, the nth term is as follows. nth term = a + (n - 1)d = 34 + (n - 1)(2) = 34 + (2n - 2) = 32 + 2n 34. 35. Each number in the sequence is 11 less than the previous number (11n). Use the first term to develop the nth term. 11 = ___ - 11(1) 11 = 22 - 11 an = 22 - 11n 36. Keep in mind that the goal is to get x by itself. 9x + 4 9x + 4 - 4 9x 9x ÷ 9 x = = = = = 22 22 - 4 18 18 ÷ 9 2 37. 38. Use properties of arithmetic to transform the given expression into one of the answer choices. [-2x - 3 - (3x - 6)]2 (-5x + 3)2 25x2 - 30x + 9 39. Use elimination by addition to solve the system of equations. Start by eliminating the y term. To do this, multiply the first equation by -3, and then add the two equations together. Next, substitute x = -5 into the first equation, and solve for y. Therefore, the solution to the system of equations is 40. Factor the equation and solve for x. . x(x - 10) = 11 x2 - 10x = 11 x2 - 10x - 11 = 0 (x + 1)(x - 11) = 0 x = -1; 11