Algebra I Semester Curriculum Study Guide

Equation Solving Fundamentals

• Core Principles:     * Be able to solve two-step, multi-step, and equations with variables on both sides.     * Include treatment for equations containing fractions and those with special solutions (no solution or infinite solutions).     * Formatting Requirements: Show all work. Each step must be on a separate line.     * Golden Rule: Whatever you do to one side of the equation, you must do to the other.     * Verification: It is a good practice to check your solutions by substitution.

• Practice Problems:     1. $-44 = -\frac{2}{3}x + 12$     2. $-12 - x = 4$     3. $\frac{3}{4}x + 6 = -8$     4. $7x - 3x - 6 = 24$     5. $5x + 3(x + 4) = 28$     6. $4x - 3(x - 2) = 21$     7. $5x - (4x - 1) = -12$     8. $9x - 5(3x - 12) = 30$     9. $80 - 9y = 6y$     10. $3x + 6 = 10 - 3(x - 2)$     11. $4(3x - 5) = 2(x - 8) - 6x$     12. $5(x - 4) = 5x + 12$     13. $3(x + 5) = 3x + 15$     14. $\frac{x}{3} = \frac{12}{2}$     15. $24w - 8 - 10w = -2(4 - 7w)$

Graphing Linear Functions

• General Instructions: Graph each equation using the method appropriate for its form. Label intercepts with ordered pairs on the graph.

• Practice Equations:     16. $y = -\frac{1}{2}x + 3$     17. $y = -2x$     18. $y = -x - 5$     19. $y = 4$     20. $y = \frac{3}{2}x$     21. $x = -5$     22. $y + 4 = 2(x - 5)$     23. $y - 2 = -\frac{1}{3}(x + 3)$     24. $2x + 3y = 6$     25. $x + y = -3$     26. $4x - 2y = 12$     27. $y - 3 = -(x + 4)$

Unit 6: Systems of Linear Equations & Inequalities

• Solving Systems by Graphing:     * Solve by graphing each line and identifying the intersection point.     * 28. System: \n        \begin{cases}\n        2x + y = 6 \\\n        y = x - 3\n        \end{cases}\n        

• Solving Systems by Substitution:     * Procedure:         1. Isolate one variable in one of the equations.         2. Substitute this expression into the other equation.         3. Solve for the remaining variable.         4. Substitute the result into either original equation to find the second variable.         5. Check solutions in both original equations.     * 29. System: \n        \begin{cases}\n        3x + y = 3 \\\n        7x + 2y = 1\n        \end{cases}\n             * 30. System: \n        \begin{cases}\n        x = y + 1 \\\n        2x - y = 8\n        \end{cases}\n        

• Solving Systems by Linear Combinations (Elimination):     * Procedure:         1. Arrange equations so like variables are aligned.         2. Aim for one variable to cancel out when adding equations.         3. Multiply one or both equations by a constant to create pairs of opposite coefficients.         4. Solve for both variables and check solutions in original equations.     * 31. System: \n        \begin{cases}\n        5x + 2y = -10 \\\n        -4x + 3y = 8\n        \end{cases}\n             * 32. System: \n        \begin{cases}\n        2x - 3y = -7 \\\n        3x + y = -5\n        \end{cases}\n        

• Special Types of Systems:     * No Solution: Occurs when both variables cancel and the remaining statement is false (e.g., $0 = 5$).     * Infinite Solutions: Occurs when both variables cancel and the remaining statement is true (e.g., $0 = 0$).     * 33. System: \n        \begin{cases}\n        -7x + 7y = 7 \\\n        2x - 2y = -18\n        \end{cases}\n             * 34. System: \n        \begin{cases}\n        4x + 4y = -8 \\\n        2x + 2y = -4\n        \end{cases}\n        

• Applications of Systems of Equations:     * Method: Identify unknowns, write a system of equations, solve, and answer the specific question.     * 35. Baseball Field Trip: A group of 40 children attended. Each received a hot dog ($2.25$) or popcorn ($1.75$). If the total bill was $83.50$, how many hot dogs and how many bags of popcorn were bought?     * 36. Pet Store Inventory: A store has cats and canaries. There are 16 heads and 50 legs. Determine the quantity of each animal.

Unit 7: Exponents & Exponential Functions

• Multiplication Properties ($x^a \cdot x^b = x^{a+b}$, $(x^a)^b = x^{ab}$):     37. $x^3 \cdot x^5$     38. $(x^3 y^5)(-yx)$     39. $(y^3)^5$     40. $(-2k^3)^5$     41. $(3x^2 y^4)^3$     42. $(x^3 y^5)(-x^4 y^6 z^3)$     43. $(5x^2 y)^4$     44. $(2a^5 b^2)^3(a^2 b)^4$

• Division Properties ($\frac{x^a}{x^b} = x^{a-b}$):     * Simplify completely; answers must contain no negative exponents.     45. $\frac{3x^5}{12x^5}$     46. $\frac{4a^3 b^8}{2ab^{10}}$     47. $\frac{(2x)^3 3x^4}{4x^2}$     48. $\frac{(3x^2 y)^4}{9x^3 y^6}$     49. $\frac{3x}{9x^2}$

• Zero & Negative Exponents Properties ($x^0 = 1$, $x^{-1} = \frac{1}{x}$):     * Simplify completely; answers must contain no negative exponents.     50. $3y^{-5}$     51. $m^2 m^{-12}$     52. $(3b^{-3})^2$     53. $\frac{1}{5y^{-x}}$     54. $\frac{m^{-2} x^2}{m^3 x^2}$     55. $\left(\frac{x^{-2}}{y^3}\right)^{-2}$

• Mixed Properties of Exponents:     56. $\frac{(3x^2 y)(-2x)^3}{(6xy^2)^4}$     57. $\frac{2x^2 y}{x^3 y^2} \cdot \frac{4x^7 y^2}{2x^3}$     58. $\frac{5x^{-2}}{3x} \cdot \frac{2y^3}{y^{10}}$     59. $\left(\frac{4x^2 y}{6xy}\right)^{-3}$

• Rational Exponents:     * Rewrite as rational exponents and evaluate. Convert decimal results to fractions if possible; otherwise, round to the nearest hundredth.     60. $\sqrt[3]{64}$     61. $(\sqrt{16})^3$     62. $(-280)^{\frac{1}{3}}$

• Simplify Rational Exponent Expressions:     63. $x^{1/3} x^{4/3}$     64. $(y^{1/6})^3$     65. $\sqrt{25m^4}$     66. $m^{10/8} m^{-3/8}$     67. $(x^{1/2} x^{1/3})^6$     68. $\sqrt[3]{x^6}$     69. $x^{-1/2}$

• Exponential Growth & Decay:     70. Business Profit: In 1990 ($t=0$), profit was $5000$ dollars. Profit increases exponentially by $12\%$ per year.         a) Write an exponential growth model.         b) Find the profit in 1995.     71. Tire Company Workforce: In 1990, there were $14,000$ employees. For 10 years, the count decreased by $4\%$ annually.         a) Write an exponential decay model.         b) Calculate the number of employees in 2000.

Unit 8: Advanced Algebra Skills

• Simplifying Radicals:     * No decimals. Leave answers as radicals in simplest form.     72. $\sqrt{44}$     73. $\sqrt{75}$     74. $\sqrt{\frac{8}{25}}$     75. $\sqrt{54}$

• Operations with Radicals:     76. $\sqrt{32} \cdot \sqrt{2}$     77. $3\sqrt{4} \cdot 6\sqrt{8}$     78. $\sqrt{12} - 2\sqrt{3}$     79. $\sqrt{20} + \sqrt{5}$

• Rationalizing the Denominator:     80. $\frac{2}{\sqrt{3}}$     81. $\frac{\sqrt{3}}{\sqrt{2}}$     82. $\frac{1}{\sqrt{20}}$     83. $\frac{3}{\sqrt{3}}$

• Adding & Subtracting Polynomials:     84. $(-7x^2 + 2x - 4) + (-2x^2 + 3x - 1)$     85. $(n^2 + 4n + 6) - (-5n^2 - 3n + 2)$

• Multiplying Polynomials (Box or Distributive Method):     86. $(-3a^2)(-4a^2 + a - 7)$     87. $(x + 9)(x - 4)$     88. $(x - 7)^2$     89. $(2x + 5)^2$     90. $(x - 3)(5x^2 + 2x + 5)$     91. $(4x + 3)(4x - 3)$

• Factoring by Greatest Common Factor (GCF):     92. $24x^3 + 18x^2$     93. $5x^3 y - 15x^2$     94. $-72w^3 - 90w$

• Factoring Polynomials (Leading Coefficient of One):     * Factor trinomials into two binomials.     95. $m^2 + 7m + 10$     96. $x^2 + 5x - 14$     97. $x^2 - 5x + 6$

• Factoring Polynomials (Leading Coefficient Greater than One):     98. $3x^2 + 17x + 10$     99. $18x^2 + 9x - 14$     100. $5x^2 - 7x + 2$

• Factoring Special Products:     * Difference of squares and perfect square trinomials.     101. $x^2 - 9$     102. $16 - 121x^2$     103. $t^2 + 10t + 25$     104. $w^2 - 16w + 64$

• Multi-Step Factoring (GCF then remaining polynomial):     105. $6y^2 - 24$     106. $18 - 2x^2$     107. $3m^3 - 27m$     108. $5x^3 + 30x^2 + 40x$     109. $45x - 20x^2$     110. $3x^4 + 12x^3 + 9x^2$

• Solving Equations using Zero Product Property:     111. $x^2 - 8 = -7x$     112. $5x^2 - 9x + 4 = 0$

• Solving Equations using Quadratic Formula:     113. $9x^2 - 25 = 0$     114. $x^2 - 4x + 2 = 0$     115. $2x^2 + 2x + 2 = 0$     116. $56x^2 + 53x = 55$

Projectile and Vertical Motion

• Vertical Motion Equations:     * Object is dropped: $h = -16t^2 + h_0$     * Object is launched or thrown: $h = -16t^2 + v_0t + h_0$     * Variables: $h$ is height, $t$ is time, $h_0$ is initial height, $v_0$ is initial velocity.

• Applied Problems:     * 117. Empire State Building Crash: On July 28, 1945, an airplane crash caused debris to fall $975$ feet. How long did it take for debris to reach the ground? (Round to the hundredth).     * 118. Cliff Diving: In July 1997, at the World Championships in Brontallo, Switzerland, a diver jumped from $92$ feet with an initial upward velocity of $5\,ft/s$. How much time does the diver have before hitting the water? (Round to the hundredth).

Optimization and Fencing Problems

119. Garden Fencing (4 sides): A rectangular garden is enclosed by $800$ meters of fencing. What are the dimensions for maximum area and what is that area? (Model with quadratic function and find the vertex).

120. Garden Fencing (3 sides): One side borders a garage; the other three sides use $90$ feet of fencing. Find maximum area and dimensions.

Pythagorean Theorem and Distance Formula

• Missing Side Problems (Round to nearest tenth):     121. Right triangle with leg $16\,in$ and hypotenuse $20\,in$. Find missing leg $x$.     122. Right triangle with legs $8\,in$ and $14\,in$. Find hypotenuse $x$.     123. Right triangle with legs $9.6\,cm$ and $12.8\,cm$. Find hypotenuse $x\,cm$.

• Right Triangle Verification:     * Do these side lengths form a right triangle? ($a^2 + b^2 = c^2$)     124. $9, 12, 16$     125. $16, 30, 34$

• Distance Between Points:     * Formula: $d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$     * Round answers to the nearest tenth.     126. Points: $(4, 8)$ and $(-5, 12)$     127. Points: $(-12, -9)$ and $(12, -2)$