Multiple Choice

1. Which of the following mathematical relationships is defined as a linear relation?

   $y = -6x + 11$

   $y = x^2 - 3x + 1$

   $y = 5x^3 + 2$

   $xy = 7$

2. A linear relation is characterized by a constant rate of change (or constant first differences) between consecutive points. Which set of $(x, y)$ values represents a linear relation?

   $(0, 10), (1, 5), (2, 2), (3, 1)$

   $(1, 5), (3, 11), (5, 17), (7, 23)$

   $(1, 2), (2, 4), (3, 8), (4, 16)$

   $(0, 0), (1, 1), (2, 4), (3, 9)$

3. Which of the following equations represents a linear relation?

   $y = \frac{3}{x} + 1$

   $xy = 20$

   $5x - 3y = 15$

   $y = x^2 - 5$

4. A linear relation is defined by a constant rate of change, also known as constant first differences. Which set of ordered pairs, showing $x$ and $y$ values, represents a linear relation?

   $x$ values are $\{-1, 0, 1, 2\}$ and $y$ values are $\{1, 0, 1, 4\}$.

   $x$ values are $\{0, 1, 2, 3\}$ and $y$ values are $\{1, 3, 6, 10\}$.

   $x$ values are $\{1, 2, 3, 4\}$ and $y$ values are $\{16, 8, 4, 2\}$.

   $x$ values are $\{-2, 0, 2, 4\}$ and $y$ values are $\{10, 7, 4, 1\}$.

5. In a linear relationship, the slope ($m$) is a critical value. Conceptually, what does the slope primarily represent?

   The starting value of the relationship (the y-intercept).

   The total value of the function at any given time.

   The specific point where the line crosses the horizontal (x) axis.

   The constant rate of change between the two variables.

6. When calculating the slope ($m$) of a straight line, we use the change in the vertical distance and the change in the horizontal distance. Which term correctly defines the ratio used for calculating the slope?

   Run over Rise ($\frac{\text{Change in } x}{\text{Change in } y}$).

   Rise over Run ($\frac{\text{Change in } y}{\text{Change in } x}$).

   Total distance traveled divided by time.

   Y-intercept divided by X-intercept.

7. Line P has a slope of $m = 8$, and Line Q has a slope of $m = 1/2$. Both lines are graphed on the same coordinate plane. What does the magnitude of the slope value tell you about Line P compared to Line Q?

   Line P increases at a slower rate than Line Q.

   Line P is less steep than Line Q.

   Line P is steeper than Line Q.

   Line P crosses the y-axis higher up than Line Q.

8. A student graphs a linear function describing the cost of buying apples, but finds that the total cost never changes regardless of the number of apples bought. If this line were graphed, what would its slope ($m$) be?

   Undefined.

   A line passing through the origin.

   Zero ($m = 0$).

   A negative number (e.g., $m = -5$).

9. A linear function models the descent of a submarine, $y = -50x + 1000$, where $y$ is the depth in meters and $x$ is the time in minutes. What does the negative slope of $-50$ specifically indicate about the submarine?

   The submarine is rising at a rate of $50$ meters per minute.

   The submarine is staying at a constant depth.

   The submarine is descending at a rate of $50$ meters per minute.

   The submarine started at a depth of $50$ meters.

10. A straight line represents the path of a bicycle rider. If the rider travels $60$ meters horizontally (run) for every $15$ meters they climb vertically (rise), what is the slope ($m$) of the path?

    $0.15$

    $60$

    $0.25$

    $4$

    $15$

11. Find the slope of the given line.

    
    -4

    1

    0

    -1/4

12. Find the slope of the given line.

    
    -2

    -1/2

    1/2

    2

13. Find the slope of the given line.

    
    3/4

    -2/3

    2/3

    5/6

14. Find the slope of the given line.

    
    1/2

    -1/2

    1

    2

15. Find the slope of the given line.

    
    -1/2

    1/2

    4

    2

16. What is the slope (gradient), $m$, of the line segment that connects the points $(2, 5)$ and $(6, 17)$?

    $3$

    $-3$

    $4$

    $12$

17. Calculate the gradient ($m$) of the line that passes through the points $(-4, 8)$ and $(4, 4)$.

    $1/2$

    $-2$

    $2$

    $-1/2$

18. What is the gradient (slope) of the straight line that passes through the points $(3, 7)$ and $(5, 15)$?

    $8$

    $4$

    $\frac{1}{4}$

    $2$

19. Calculate the slope ($m$) of the linear relation passing through the points $(-2, 10)$ and $(4, -8)$.

    $-3$

    $-9$

    $\frac{1}{3}$

    $3$

20. A straight line is drawn on a coordinate grid. The line passes through the points $(0, 3)$ and $(1, 5)$. Which equation correctly models the relationship between $x$ and $y$?

    $y = 2x + 3$

    $y = 5x + 3$

    $y = 3x + 2$

    $y = 2x - 3$

21. A linear relation is graphed, passing through the points $(-2, 0)$ and $(0, -1)$. What is the correct equation for this line?

    $y = \frac{1}{2}x - 1$

    $y = -\frac{1}{2}x - 1$

    $y = -2x - 1$

    $y = -\frac{1}{2}x + 1$

22. In which quadrant does the point $P(-5, 9)$ lie?

    Quadrant III

    Quadrant I

    Quadrant II

    Quadrant IV

23. A point $R$ lies on the y-axis and is 7 units below the origin. What are the coordinates of point $R$?

    $(7, 0)$

    $(0, -7)$

    $(-7, 0)$

    $(0, 7)$

24. For the ordered pair $(-11, 4)$, what is the term used to describe the first coordinate, $-11$?

    Origin

    Ordinate

    Abscissa

    Quadrant

25. If a point $T(a, b)$ is located in Quadrant IV, which statement correctly describes the signs of the coordinates $a$ and $b$?

    $a < 0$ and $b < 0$

    $a > 0$ and $b > 0$

    $a < 0$ and $b > 0$

    $a > 0$ and $b < 0$

26. Which algebraic expression represents the phrase "seven more than four times a number, $x$"?

    $7(4x)$

    $4(x + 7)$

    $7x + 4$

    $4x + 7$

27. Translate the following sentence into an algebraic equation: "When 15 is subtracted from half of a number, $n$, the result is 30."

    $15 - \frac{n}{2} = 30$

    $\frac{15}{n} - 2 = 30$

    $2n - 15 = 30$

    $\frac{n}{2} - 15 = 30$

28. Write the algebraic expression for the phrase "the square of the difference between a number $y$ and 5."

    $(y - 5)^2$

    $5^2 - y^2$

    $y - 5^2$

    $y^2 - 5^2$

29. An algebraic equation is used to model the relationship: "Three times a number, $p$, increased by 11 is the same as 56."

    $3p + 11 = 56$

    $3p - 11 = 56$

    $3(p + 11) = 56$

    $11p + 3 = 56$

30. Simplify the expression: $3(2x - y) - (4x + 5y)$.

    $2x + 8y$

    $2x - 8y$

    $10x - 8y$

    $2x + 2y$

31. Simplify the algebraic expression: $5a - [2a + 3(a - 4)]$.

    $2a - 12$

    $12a - 4$

    $-12$

    $12$

32. Combine like terms to simplify the expression: $(4y^2 + 7y - 1) - (y^2 - 3y + 6)$.

    $3y^2 + 10y - 7$

    $3y^2 + 4y - 7$

    $5y^2 + 10y + 5$

    $3y^2 + 4y + 5$

33. Simplify the following expression: $\frac{1}{2}(4m - 6n) + 5m - 3(n + m)$.

    $10m - 3n$

    $4m - 6n$

    $4m$

    $10m - 6n$

34. Identify the Greatest Common Monomial Factor (GCMF) of the expression $12x^3 - 18x^2$.

    $6$

    $6x^3$

    $3x^2$

    $6x^2$

35. What is the Greatest Common Monomial Factor (GCMF) for the terms in the algebraic expression $-20a^4b^3 + 15a^2b^5$?

    $5a^2b^3$

    $5a^4b^3$

    $5ab$

    $5a^2b^5$

36. Factor the expression $9m^5n^2 - 6m^3n^3 + 3m^2n^4$ completely by extracting the Greatest Common Monomial Factor.

    $3m^2n^2(3m^3 - 2mn + n^2)$

    $3m^2n^2(3m^3 - 2mn + n^4)$

    $3m^2n^2(3m^3 - 2m^3n + n^2)$

    $m^2n^2(9m^3 - 6mn + 3n^2)$

37. Determine the Greatest Common Monomial Factor (GCMF) of $24y^6z^3$ and $36y^4z^5$.

    $12y^4z^3$

    $4y^4z^3$

    $6y^4z^3$

    $12y^6z^5$

38. Find the Greatest Common Monomial Factor (GCMF) of the expression $12x^2 + 18x$.

    $12x$

    $6x$

    $6x^2$

    $3x$

39. Determine the fully factored form of the expression $-10a^3b^2 + 15a^2b^3$.

    $5a^3b^3(-2 + 3)$

    $5ab(-2a^2b + 3ab^2)$

    $5a^2b^2(-2a - 3b)$

    $5a^2b^2(-2a + 3b)$

40. Identify the Greatest Common Monomial Factor (GCMF) of the polynomial $6k^5m^3 - 12k^4m^4 + 3k^2m^5$.

    $6k^2m^3$

    $3k^2m^3$

    $3k^2m^5$

    $3k^4m^3$

41. If the expression $24y^7 - 8y^4 + 16y^2$ is factored completely using the GCMF, which trinomial remains inside the parentheses?

    $3y^3 - y^2 + 2$

    $3y^5 - y^2 + 2$

    $3y^5 + y^2 + 2$

    $24y^5 - 8y^2 + 16$

42. Simplify the expression $-4(y - 7)$ by applying the distributive property.

    $-4y + 7$

    $4y - 28$

    $-4y + 28$

    $-4y - 28$

43. Determine the expanded form of the product $(x + 3)(x - 2)$.

    $x^2 - 6$

    $x^2 - x - 6$

    $x^2 + x + 6$

    $x^2 + x - 6$

44. Expand the expression $(x + 5)(x + 3)$ using the FOIL method.

    $x^2 + 15$

    $x^2 + 15x + 15$

    $x^2 + 8x + 15$

    $x^2 + 8x + 8$

45. Determine the expanded form of the binomial product $(y - 7)(y + 2)$.

    $y^2 + 5y - 14$

    $y^2 - 5y - 14$

    $y^2 - 5y + 14$

    $y^2 - 9y - 14$

46. What is the result of expanding the expression $(2m - 1)(3m + 4)$?

    $5m^2 + 5m - 4$

    $6m^2 + 5m - 4$

    $6m^2 + 11m - 4$

    $6m^2 - 5m - 4$

47. Expand and simplify the expression $(4k - 9)(4k + 9)$.

    $16k^2 - 81$

    $16k^2 + 72k - 81$

    $8k^2 - 81$

    $16k^2 + 81$

48. What is the value of $x$ that satisfies the equation $5x - 8 + 2x = 20$?

    $x = 12/7$

    $x = 28$

    $x = 2.5$

    $x = 4$

49. Solve for $x$: $3(4x - 5) = 33$.

    $x = 1.5$

    $x = 4$

    $x = 19/6$

    $x = 3$

50. Determine the solution to the linear equation $7x - 12 = 3x + 8$.

    $x = -1$

    $x = 0.4$

    $x = 5$

    $x = 2$

51. Solve the following equation for $x$: $2(3x - 4) + 10 = 5x + 15$.

    $x = 19$

    $x = -3$

    $x = 13$

    $x = 17$

52. Write the equation of the line

    
    y=-1/2x+3

    y=2x+3

    y=1/2x+3

    y=2x-3

53. Write the equation of the line

    
    y=-2/5x-4

    y=1/10x-5

    y=10x-5

    y=2/5x-5

54. Write the equation of the line

    
    y=1/3x+3

    y=3x+3

    y=-3x+3

    y=-1/3x+3

Numerical

55. What is the degree (highest exponent) of the independent variable $x$ in the equation $y = \frac{1}{2}x + 9$?

56. Consider the following list of equations. How many of these equations represent a linear relation (meaning the highest degree of the variables is 1)? I. $y = 5x - 4$ II. $y = 3x^2 + 1$ III. $x + 2y = 10$ IV. $y = \frac{1}{x}$

57. A construction crew is building a roof. For every $25$ feet the roof extends horizontally (the run), it rises $2$ feet vertically (the rise). What is the slope (gradient) of the roof? (Express your answer as a decimal).