# Algebra 2 PreAP. Name Period

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2 Solving Equations Addition Property of Equality For any numbers a, b, and c, if a = b, then a + c = b +c. Subtraction Property of Equality For any numbers a, b, and c, if a = b, then a c = b c. Multiplication Property of Equality For any numbers a, b, and c, if a = b, then ac = bc. Division Property of Equality For any numbers a, b, and c, with c 0, if a = b, then a = b. c c Example 1 Solve: 3 ½ p = 1 ½ 7 2 p = p p = Check: 3 ½ 7 = 1 ½ = Original equation Rewrite each mixed number as an improper fraction Multiply each side by the reciprocal of 7/2. Simplify Substitute solution for variable Rewrite each mixed number as an improper fraction Left Hand Side = Right Hand Side LHS = RHS correct Example 2 Solve: -5n = 60 Original equation -5n = 60 Divide both sides by -5 or -5-5 multiply both sides by -1/5 n = -12 Simplify Check: -5 (-12) = 60 Substitute solution for variable 60 = 60 Left Hand Side = Right Hand Side LHS = RHS correct Solving Equations with the Variable on Each Side To solve an equation with the same variable on each side, first use the Addition or the Subtraction Property of Equality to write an equivalent equation that has the variable on just one side of the equation. Then use the Multiplication or Division Property of Equality to solve the equation. Example 1 Solve: 5y 8 = 3 y y 8 3y = 3 y y 2y 8 = 12 2y = y = 20 2y = y = 10 Check: 5y 8 = 3 y (10) 8 = 3(10) = = 42 LHS = RHS correct Example 2 Solve: -11 3y = 8y y + 3y = 8y y -11 = 11y = 11y = 11y -12 = 11y = y = y Check: -11 3y = 8y = = = = LHS = RHS correct Example 3 Solve: 4 (2a 1) = -10 (a 5 ) 8a 4 = -10a a a = -10 a a 18a 4 = a = a = 54 18a = a = 3 Check: 4 (2a 1) = -10 (a 5 ) 4 (2(3) 1) = -10 (3 5 ) 4 ( 6 1 ) = -10(-2) 4 (5) = = 20 LHS = RHS correct

3 Exercises: Solve each equation. Then check your solution = y 8 2. w ½ = = b h = m = p = h = ½ m = t = b = 5b y 2 y = 3y x + 2 = 2x n 8 = 3n x = 2.1 x s = 8.8s ½ b + 4 = 1 8 b ¾ k 5 = ¼ k p = 4p 1 3

4 19. -3(x + 5) = 3(x - 1) 20. 2(7 + 3t) = -t 21. 3(a + 1) 5 = 3a 2 Solving Equations and Formulas Solve for variables: sometimes you may want to solve an equation such as V = lwh for one of its variables. For example, if you know the values of V, w, and h, then the equation l = V is more useful for finding the value of l. wh Example 1 Solve 2x 4 y = 8 for y. 2x 4y = 8 2x 4y 2x = 8 2x -4y = 8 2x -4y = 8 2x -4-4 y = 8-2x or 2x Example 2 Solve 3m n = km 8 3m n = km 8 3m n km = km 8 km 3m n km = -8 3m n km + n = -8 + n 3m km = -8 + n m (3 k ) = -8 + n m (3 k ) = -8 + n 3 k 3 k m = -8 + n, or n 8 3 k 3 k Exercises: Solve each equation or formula for the variable specified. Since division by 0 is undefined, 3 k 0, or k ax b = c for x x + 1 = y for x 24. (x + f) + 2 = j for x 25. xy + z = 9 for y 26. x( 4 k ) = p for k 27. 7x + 3y = m for y 28. xy + xz = 6 + a for x 4

5 Describe Number Patterns Write Equations: Sometimes a pattern can lead to a general rule that can be written as an equation. Example: Suppose you purchased a number of packages of blank CDs. If each package contains 3 CDs, you could make a chart to show the relationship between the number of packages of compact disks and the number of disks purchased. Use x for the number of packages and y for the number of compact disks. Make a table of ordered pairs for several points of the graph. Number of packages Number of CDs The difference in the x values is 1, and the difference in the y values is 3. This pattern shows that y is always three times x. This suggests the relation y = 3x. Since the relation is also a function, we can write the equation in functional notation as f(x) = 3x. Exercises: 29. Write an equation for the function in functional notation. Then complete the table. x y Write an equation for the function in functional notation. Then complete the table. Example 1 x y Write an equation in functional notation. 32. Write an equation in functional notation. 5

6 Equations of Linear Function Standard Form Slope-Intercept Form Point-Slope Form Ax + By = C y = mx + b, where m is the given slope and b is the y-intercept y- y 1 = m (x x 1 ), where m is the given slope and (x 1, y 1 ) is the given point Example 1: Write an equation of a line in standard form whose slope is -4 and whose y-intercept is 3. y = mx + b y = -4x x +4x 4x + y = 3 Example 2: Graph 3x 4y = 8 3x 4y = 8 Original equation -4y = - 3x + 8 Subtract 3x from each side -4y = - 3x + 8 Divide each side by y = ¾ x 2 Simplify The y-intercept of y = 3/4x 2 is -2 and the slope is ¾. So graph the point (0, -2). From this point, move up 3 units and right 4 units. Draw a line passing through both points. Exercises: Write an equation of the line in Standard Form with the given information. 33. Slope: 8, y-intercept Slope: -2, point (5, 3) 35. Slope: -1, y-intercept -7 Write an equation of the line in Standard Form in each graph Graph each equation x y = x + y = x + y = -1 6

7 Graphing Systems of Equations Solve by Graphing One method of solving a system of equations is to graph the equations on the same coordinate plane. Example: Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. a. x + y = 2 x y = 4 The graphs intersect. Therefore, there is one solution. The point (3, -1) seems to lie on both lines. Check this estimate by replacing x with 3 and y with -1 in each equation. x + y = (-1) = 2 x y = 4 3 (-1) = or 4 The solution is (3,-1). b. y = 2x + 1 2y = 4x + 2 The graphs coincide. Therefore there are infinitely many solutions. Exercises: Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. 42. y = x = y = ½ x 3x y = -1 2x + y = 1 x + y = x + y = x + 2y = y = -4x + 4 2x y = -2 3x + 2y = -4 y = -2x + 2 7

8 Solving Systems of Equations by Substitution Example 1: use substitution to solve they system of equations. y = 2x 4x y = -4 Substitute 2x for y in the second equation. 4x y = -4 second equation 4x 2x = -4 y = 2x 2x = -4 combine like terms x = -2 Divide each side by 2 and simplify. Use y = 2x to find the value of y. y = 2x First equation y = 2(-2) x = -2 y = -4 simplify The solution is (-2,-4). Example 2: Solve for one variable, then substitute. x + 3y = 7 2x 4y = -6 Solve the first equation for x since the coefficient of x is 1. x + 3y = 7 First equation x + 3y 3y = 7 3y Subtract 3y from each side x = 7 3y Simplify Find the value of y by substituting 7 3y for x in the second equation. 2x 4y = -6 Second equation 2(7-3y) 4y = -6 x = 7 3y 14-6 y 4y = -6 Distributive Property 14-10y = -6 Combine like terms y 14 = Subtract 14 from each side. -10y = -20 Simplify. y = 2 Divide each side by -10 and simplify. Use y = 2 to find the value of x. x = 7-3y x = 7 3(2) x = 1 The solution is (1, 2). Exercises: Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions. 48. y = 4x 49. x = 2y 50. x = 2y - 3 3x y = 1 y = x 2 x = 2y + 4 8

9 Elimination Using Addition and Subtraction Elimination Using Addition: In systems of equations in which the coefficients of the x or y terms are additive inverses, solve the system by adding the equations. Because one of the variables is eliminated, this method is called elimination. Example 1: Use addition to solve the system of equations x 3y = 7 3x + 3y = 9 Write the equations in column form and add to eliminate y. x 3y = 7 (+) 3x+ 3y = 9 4x = 16 4x = x = 4 Substitute 4 for x either equation and solve for y. 4x 3 y = 7 4 3y 4 = 7 4-3y = y = -1 The solution is (4, -1). Example 2: The sum of two numbers is 70 and their difference is 24. Find the numbers. Let x represent one number and y represent the other number. x + y = 70 (+) x y = 24 2x = 94 2x = x = 47 Substitute 47 for x in either equation y = y 47 = y = 23 The numbers are 47 and 23. Exercises: Use elimination to solve each system of equations. 51. x + y = m 3n = a b = -9 x y = 2 m + 3n = -11-3a 2b = 0 9

10 Multiplying a Polynomial by a Monomial Product of Monomial and Polynomial: The Distributive Property can be used to multiply a polynomial by a monomial. You can multiply horizontally or vertically. Sometimes multiplying results in like terms. The products can be simplified by combining like terms. Example 1: Find -3x 2 (4x 2 + 6x 8). -3x 2 (4x 2 + 6x 8) = -3x 2 (4x 2 ) + (-3x 2 )(6x)-(-3x 2 )(8) = -12x 4 + (-18x 3 ) (-24x 2 ) = -12x 4-18x x 2 Example 2: Simplify -2(4x 2 + 5x) x (x 2 + 6x) -2(4x 2 + 5x) x (x 2 + 6x) = -2(4x 2 ) + (-2)(5x) + (-x)(x 2 ) + (-x)(6x) = -8x 2 + (-10x) + (-x 3 ) + (-6x 2 ) = (-x 3 ) + [ -8x 2 + (-6x 2 )] + (-10x) = -x 3 14x 2 10x Exercises: Find each product. 54. x(5x + x 2 ) 55. x (4x 2 + 3x + 2) xy(2y + 4x 2 ) g (g 2-2g +2 ) 58. 3x (x 4 + x 3 +x 2 ) x (2x 3-2x + 3) Factoring Using the Greatest Common Factor Example 1: Use GCF to factor 12mn + 80m 2 Find the GCF of 12mn and 80m 2 12mn = m n 80 m 2 = m m GCF = 2 2 m or 4m Write each term as the product of the GCF and its remaining factors. 12mn + 80m 2 = 4m (3 n) + 4m (2 2 5 m) = 4m (3n) + 4m (20m) = 4m (3n + 20 m) 12mn + 80m 2 = 4m (3n + 20 m) Example 2: Factor 6ax + 3ay + 2bx + by by grouping. 6ax + 3ay + 2bx + by = (6ax + 3ay) + (2bx + by) = 3a (2x + y) + b (2x + y) = (3a + b)( 2x + y) Check using the FOIL method. (3a + b )(2x + y) = 3a(2x) + (3a)(y) + (b)(2x) + (b)(y) = 6ax + 3ay + 2bx + by Exercises: Factor each polynomial x + 48 y mn 2 + m 2 n 6n 62. q 4 18q q 63. 9x 2 3x 64. 4m + 6n 8mn s 3 15s c 3 42c 5 49c p 2 11p p y 3 28y 2 + y 10

11 Multiplying Polynomials Multiply Binomials: To multiply two binomials, you can apply the Distributive Property twice. You can use FOIL (First, Outer, Inner and Last) method. Example 1: Find (x + 3)(x 4) (x + 3)(x 4) = x (x 4) + 3 (x 4) = (x)(x) + x (-4) + 3(x) + 3(-4) = x 2 4x + 3x 12 = x 2 x 12 Example 2: Find (x 2) (x + 5) using FOIL method. (x 2) (x + 5) First Outer Inner Last = (x)(x) + (x)(5) +(-2) (x) +(-2)(5) = x 2 + 5x + (-2x) 10 = x 2 + 3x 10 Exercises: Find each product. 69. (x + 2)(x + 3) 70. (x 4)(x +1) 71. (x 6)(x 2) 72. (p 4)(p + 2) 73. (y + 5)(y + 2) 74. (2x 1)(x + 5) 75. (3n 4)(3n 4) 76. (8m 2 )(8m + 2) 77. (k + 4)(5k 1 ) 11

12 Factoring Trinomials: x 2 +bx +c Factor x 2 +bx +c : To factor a trinomial of the form x 2 +bx +c, find two integers m and n, whose sum is equal to b and whose product is equal to c. Example 1: Factor each trinomial. a. x 2 + 7x + 10 In this trinomial, b = 7 and c = 10. Factors of 10 Sum of Factors 1, , 5 7 x 2 + 7x + 10 = (x + 5)(x + 2) b. x 2-8x + 7 In this trinomial, b = -8 and c = 7. Notice that m + n is negative and mn is positive, so m and n are both negative. Since -7 + (-1) = -8 and (-7)(-1) = 7, m = -7 and n = -1. x 2 8x + 7 = (x 7)(x 1) Example 2: Factor x 2 + 6x 16 In this trinomial, b = 6 and c = -16. This means m + n is positive and mn is negative. Make a list of the factors of -16, where one factor of each pair is positive. Factors of -16 Sum of Factors 1, , , , 8 6 Therefore, m = -2 and n = 8. x 2 + 6x 16 = (x 2)(x + 8) Exercises: Factor each trinomial. 78. x 2 + 4x m m r 2 3r x 2 x x 2 4x x 2 22x c 2 4c p 2 16p x + x x 2 + 6x a 2 + 8a y 2 7y x 2 2x y 2 +14y m 2 + 9m

13 Factoring Trinomials: ax 2 +bx +c Factor ax 2 +bx +c : To factor a trinomial of the form ax 2 +bx +c, find two integers m and n, whose sum is equal to b and whose product is equal to ac. If there are no integers that satisfy these requirements, the polynomial is called a prime polynomial. Example 1: Factor 2x 2 +15x +18. In this example, a = 2, b =15, and c = 18. You need to find two numbers whose sum is 15 and whose product is 2 18 or 36. Make a list of the factors of 36 and look for the pair of factors whose sum is 15. Factors of 36 Sum of Factors 1, , , Use the pattern ax 2 +mx + nx +c with a= 2, m= 3, n = 12 and c = 18. 2x 2 +15x +18 = 2x 2 +3x + 12x +18 = (2x 2 +3x) + (12x +18) = x(2x + 3) + 6(2x + 3) = (x + 6) (2x + 3) Example 2: Factor 3x 2 3x 18 Note that the GCF of the terms 3x 2, 3x, and 18 is 3. First factor out this GCF. 3x 2 3x 18 = 3(x 2 x 6). Now factor x 2 x 6. Since a = 1, find the two factors of -6 whose sum is -1. Factors of -6 Sum of Factors 1, , 6 5-2, 3 1 2, -3-1 Now use the pattern (x + m) (x + n) with m = 2 and n = -3. x 2 x 6 = (x + 2) (x 3) The complete factorization is 3x 2 3x 18 = 3(x + 2) (x 3). Exercises: Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime x 2 3x m 2 8m r 2 8r x 2 + 5x x 2 + 2x x 2 27x a 2 +5a y 2 + 9x c 2 +19c 21 13

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