Name: _______________

College Prep Algebra II Summer Packet Please complete and bring this packet to class on the first day of school. Show ALL work! There will be a test soon after. Remember: When simplifying fractions the numerator and denominator should not have any common factors.

15 3 5   85 3 17

Simplify and leave in fraction form (no decimals). 1.

12  21

Remember: When multiplying fractions you multiply the numerators and then the denominators. 2 15 30 3    (Always Simplify!) 1 1 5 8 40 4   3 6 When adding or subtracting fractions you must have a common denominator. 12 1 Multiply one or both of the fractions by an equivalent to one that is also the    least common multiple of the denominators. 32 6 Once you have a common denominator add or subtract the numerators only. 2 1 3 1    6 6 6 2 Perform the indicated operation. Leave all fractions in simplified terms (no decimals). 2.

2 9   3 12

3.

1 5   4 6

4.

9 3   10 5

1

Remember: Step 1: Multiply by using the distributive property Step 2: Add like terms on each side of the equal sign Step 3: Combine like terms by adding or subtracting Step 4: Divide both sides by 3 Solve for the variable. 5. 7  3(4  3x)

6. 3m  2  4m 11

7. 7( x  2)  8  3( x  4)  2

8.

x 1  5 3

9.

6  2 k

2

5( x  6)  8  2( x  3)  7 5x  30  8  2 x  6  7 5x  38  2 x 13 3x  51 x  17

Remember: Pythagorean’s Theorem can be used to find missing side lengths in a right triangle. a 2  b2  c 2

OR

leg  leg  hyp 2 2

2

10. A skateboard ramp is 5 feet long. The bottom of the ramp is 4 feet long. About how tall is the ramp? Explain how you got your answer.

5 ft

4 ft

11. Determine the length of the rafter x for the house diagrammed below. x

7 10

Remember: The rules of exponents are: Rule name

Rule

Example

Product Rules

a n · a m = a n+m

23 · 24 = 23+4 = 128

Quotient Rules

a n / a m = a n-m

25 / 23 = 25-3 = 4

Power Rules

(bn)m = bn·m

(23)2 = 23·2 = 64

b-n = 1 / bn

2-3 = 1/23 = 0.125

Simplify.

Negative Exponents

12.

 a6b4  ab6 

13.

3

60 x3 y 5 2 xy 3

3 14. 4 x  2 x  4 

15.

 x  x  8

17.

x7 y3 x3 y 4

19.

x 4 x 7

16.

6 x2 12 x 3

18.

 2x y 

20.

121 36

21.  81

22.

36x 2

23.

4

5 3

4

4x4 y 2 16 x 2 y

5

Remember: Be sure to READ the problem and perform the indicated operation. ( x  6)( x  3)

For example, multiplying two binomials: Step 1: Multiply by using the distributive property (FOIL)

F O I x2  3x  18

Step 2: Combine like terms Simplify. 24.

 5x

25.

 3x

2

2

 2 x  1   3x 2  3x  18

 6x  7    x2  2x  4

26. 3a 8a  10 

27. g  4 8g  3k 

28.

 x  1 2 x  4

29.

 2 x  3 2 x  3

30.

 4 x  7  2 x  2

31.

 3x  4 

x 2  3x  6 x  18

2

5

L

Remember: The Greatest Common Factor (GCF) is the largest term that can be divided into each of the terms. For example: Step 1: Find the GCF of the integers and the variables Step 2: Divide the GCF into each term and write its quotient Step 3: To check your work, distribute the GCF over the polynomial. You should always get the original polynomial.

7 x3 y 2  21x 2 y  14 xy3 7 xy(______  ______  ______) 7 xy( x2 y  3x  2 y 2 )

Factor out the Greatest Common Factor. 32.

30 x 2  12 x

33. 15xy 2  21x 3 y 5

34. 12 x2 y 4  36 x5 y3  8x3 y 2

Remember: When factoring polynomials you are looking for the pair of factors of the third term that add up to the second term. Don’t forget – always look for a GCF first! x2  3x  28

For example, to factor trinomials with a squared term: Step 1: Find the factor pairs of 28 + -1 2 4

28 14 7

Step 2: Chose the pair that add up to 3 . Watch your signs!  x  4 x  7  Step 3: To check your work, multiply the terms using the FOIL method. You should always get the original polynomial. Factor into two binomials 35. x 2  5x  4

36. x 2  6 x  5

37. 2 x2  14 x  24

38. x 2  x  20

6

39. 2 x2  7 x  4

40. 3x2  17 x  10

Solve: 41.

x2  7 x  8  0

42. 2 x2  7 x  3  0

Remember: There can be one, none or infinitely many solutions to a system of equations. When graphing linear equations written in y  mx  b form, m represents the slope and b represents the intercept. 2 For example, a linear equation that is solved for y: y  x4 3 Step 1: Plot the y-intercept by identifying ‘b’ b  4 so plot the point  0, 4  m

Step 2: Identify your slope ‘m’

y-

2 3

rise so, from  0, 4  go up two and run then to the right three plotting a second point of the line at  3, 2  . Plot at least three points when graphing a line. Step 3: From the y-intercept plot points using the slope. Don’t forget slope is

Step 4: Write solution as an ordered pair

43. Solve the system of equations by graphing. y  3x  4 y  x  8

7

44. Solve the system of equations by graphing. y  3x  1 y  3x  5

Remember: When solving a system by elimination you first either add or subtract the equations to eliminate one of the variables.

x  y  10 x  y  6 Step 1: Add the equations to eliminate the y variable 2x  4 Step 2: Solve for x x2 Step 3: Substitute the value of x into either of the original equations in order to solve for y. 2  y  10 y 8

 2,8

Step 4: Write the solution as an ordered pair

When solving a system by substitution one equation must be written in terms of one of the variables. The equation is then substituted into the second equation to eliminate one of the variables.

x  y 6 y  x  10 Step 1: Substitute the first equation into the second to eliminate the x variable

y   y  6   10 2 y  6  10 2 y  16

Step 2: Solve for y

y 8 Step 3: Substitute the value of y into either of the original equations in order to solve for x. x  86 x2 Step 4: Write the solution as an ordered pair  2,8 8

Solve the following systems of equations: x y 3 45. x y 7

46.

47.

x  4y  8  2 x  6 y  8

3x  y  1 2 x  2 y  2

48. Circle the system graphed below that has the solution  5, 4  . a.

b.

c.

9

d.

Remember: When finding the slope and y-intercept form from an equation, the equation first must be in y  mx  b form, with m being the slope and b the y-intercept. 49. Find the slope of the line y  2 x  4.

50. Find the slope of the line 3 y  5x.

51. Find the slope of the line 2 x  3 y  6.

52. Find the y-intercept of the line y 

2x 3  . 5 5

53. Find the y-intercept of the line 5x  4 y  3.

54. In slope-intercept form, the y-intercept is represented by: a) m

b) b

c) x

d) y

e) y1

55. Which of these points is on the line 3x  y  7?

a)

3, 1

b) 3,1

c)

1,1

d )  4,5 

10

e)

0,0 

Remember: When finding the equation of a line, first you need a slope. If you are given two points, use the y y slope formula, m  2 1 , to find the slope. Then use the point-slope formula, x2  x1

y  y1  m  x  x1  to find the equation of the line. Substitute the slope in for m and one of the

points into x1 and y1

56. Write the equation of the line with a slope of -2 and passing through the point  0,1 .

57. Write the equation of the line passing through the points  6, 1 and  8, 0  .

58. Graph the line x  5 .

11

Remember: Parallel lines have the same slope. Perpendicular lines have opposite sign and reciprocal slopes, 2 2 for example and - . 3 3

59. What is the slope of a line parallel to y 

1 x3 ? 2

60. What is the slope of a line perpendicular to y 

3 x6? 5

61. Which pair of equations represents perpendicular lines? a. y 

1 1 x  1 and y   x  3 4 4

b. y 

1 x  2 and y  3x  5 3

d. y  7 x  4 and y  7 x  6

c. y  x and y  2 x

62. When graphing a system of parallel lines, the number of solutions that will result is:

a. 0

b. 1

c. 2

d. infinitely many

63. When a system of equations has infinitely many solutions, the lines are __________.

a. perpendicular

b. the same line

c. parallel

12

d. intersecting