Mathematics Problems: Equations, Functions, Algebra, and Graphs, Lecture notes of Linear Algebra

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1. Which equation is equivalent to x + 2 y = 6?
   1. y = − x + 6
   2. y = − x + 6
   3. y = − x + 3
   4. y = − x + 3
2. Which table represents a function?

3. Write the equation for the line shown in the accompanying graph. Answer: y = x +
4. Solve the following system of equations algebraically and check: 2 x + 3 y = 11 5 x − 2 y = − The solution is: x = , y =.
5. Solve the following system of equations algebraically and check: 3 x − 5 y = − 2 x − 3 y = − Answer: x = and y =

Figure 1 Use the diagram of intersecting lines to answer the question.

8. [Refer to figure 1] In the diagram, the measure of ∠ 1 = °
9. Factored completely, the expression 3 x^3 – 33 x^2 + 90 x is equivalent to
   1. 3 x ( x^2 – 33 x + 90)
   2. 3 x ( x^2 – 11 x + 30)
   3. 3 x ( x + 5)( x + 6)
   4. 3 x ( x – 5)( x – 6)
10. Express x^2 − 5 x − 24 as the product of two binomials.
    1. ( x − 8)( x + 3)
    2. ( x + 8)( x − 3)
    3. ( x + 6)( x + 4)
    4. ( x − 6)( x − 4)
11. The length of the hypotenuse of a right triangle is 34 inches and the length of one of its legs is 16 inches. What is the length, in inches, of the other leg of this right triangle?
12. 16 3. 25
13. 18 4. 30
14. Express the product (2 x − 1)(3 x + 4) as a trinomial:
    1. 5 x^2 + 9 x − 4
    2. 5 x − 4
    3. 6 x^2 − 5 x − 4
    4. 6 x^2 + 5 x −
       13. Factor: x^2 − x − 6
       14. ( x − 3) ( x + 2)
       15. ( x + 3) ( x − 2)
       16. ( x − 1) ( x − 6)
       17. ( x − 6) ( x − 1)
       18. Solve for x : 4(2 x − 1) = 2 x + 35
       19. 3.9 3. 6
       20. 6.5 4. 39
       21. What are the factors of x^2 − 10 x − 24?
       22. ( x − 4)( x + 6)
       23. ( x − 4)( x − 6)
       24. ( x − 12)( x + 2)
       25. ( x + 12)( x − 2)
       26. The product of 3 x^2 y and −4 xy^3 is
       27. −12 x^3 y^4
       28. 12 x^3 y^4
       29. −12 x^2 y^3
       30. 12 x^2 y^3

17. The expression is equivalent to
    1. 5 k^2 − 3 k + 1
    2. 5 k^2 − 3 k
    3. 15 k^3 − 9 k^2
    4. 3 k
18. The sides of a triangle measure 9, 15, and 18. If the shortest side of a similar triangle measures 6, find the length of the longest side of this triangle.
19. 5 3. 12
20. 10 4. 15
21. The sum of 2 xy and 17 xy is
    1. 19 x^2 y^2
    2. 19 xy
    3. 34 xy
    4. 15 xy
22. The line 3 x - 2 y = 12 has
    1. a slope of and a y -intercept of -
    2. a slope of - and a y -intercept of 6
    3. a slope of 3 and a y -intercept of -
    4. a slope of -3 and a y -intercept of -
23. Factor: y^2 − 100
    1. ( y − 10)^2
    2. ( y − 10)( y − 10)
    3. ( y + 10)( y + 10)
    4. ( y + 10)( y − 10)
24. What is the sum of 6 × 10^3 and 3 ×10^2?
    1. 6.3 ×10^3
    2. 9 ×10^5
    3. 9 × 10^6
    4. 18 ×10^5
       23. An equation whose graph has a slope of −2 and a y -intercept of 3 is
       24. x = −2 y + 3
       25. y = −2 x + 3
       26. x = 3 y − 2
       27. y = 3 x − 2
       28. What is the slope of the line y = x + 1?
       29. 1
       30. 2
       33. What is the slope of the line whose equation is 3 x + y = 4?
       34. 3 x 3. 3
       35. -3 4. 4
       36. Find the coordinates of the image of (−3, 4) under the transformation T (^) −2, 3.
       37. (1, −1) 3. (5, −7)
       38. (−1, 1) 4. (−5, 7)
       39. Find the sum of 4 x^2 − 3 x + 6 and −7 x^2 + 3 x + 6
       40. 7 x^2 + 6 x + 12
       41. −3 x^2 + 6 x + 12
       42. 3 x^2 − 12
       43. −3 x^2 + 12
       44. Use the function f( x ) = 2 x + 3 to find the value of f(4). f(4) =
       45. Express (4 x − 5)(6 x + 5) as a trinomial.
       46. 24 x^2 − 10 x − 25
       47. 24 x^2 − 50 x − 25
       48. 24 x^2 − 25
       49. −24 x^2 + 25

Figure 2 Two parallel lines are cut by a transversal, forming eight angles as shown in the diagram.

32. [Refer to figure 2] ∠ 3 and ∠ are alternate interior angles. ∠ and ∠ 5 are alternate interior angles.
33. Which polynomial is the quotient of ?
34. 2 x^2 + 3 x + 1
35. 2 x^2 + 3 x
36. 2 x + 3
37. 6 x^2 + 9 x
38. What is the slope of line l shown in the accompanying diagram?
41. −
42. −

35. If the number of molecules in 1 mole of a substance is 6.02 × 10^23 , then the number of molecules in 100 moles is
    1. 6.02 × 1021
    2. 6.02 × 10^22
    3. 6.02 × 10^24
    4. 6.02 × 10^25
36. The expression ( x^2 − 5 x − 2) − (−6 x^2 − 7 x − 3) is equivalent to
    1. 7 x^2 − 12 x − 5
    2. 7 x^2 − 2 x + 1
    3. 7 x^2 + 2 x + 1
    4. 7 x^2 + 2 x − 5
37. Write the equation for the line shown in the accompanying graph. y = x +

49. Which rule describes the ordered pairs in this table?
    1. y = 3 x + 2
    2. y = 2 x + 1
    3. x = y – 4
    4. y = x + 2
50. The expression (2 x^2 + 6 x + 5) − (6 x^2 + 3 x + 5) is equivalent to
    1. −4 x^2 + 3 x
    2. 4 x^2 − 3 x
    3. −4 x^2 − 3 x + 10
    4. 4 x^2 + 3 x − 10
51. Find x : -4 x - 8 = - x =
    52. Simplify and solve for a : -3( a - 2) + 1 = 28. Answer: a =
    53. Find the value of x : 5 + 2 x = x - 3. Answer: x =
    54. What is expressed in simplest radical form?
    55. 16
    56. 4
    57. 4
    58. 2
    59. What is expressed in simplest radical form?
    60. 2
    61. 3
    62. 6
    63. 8