Multiple Choice

1. A line passes through the points $(a, 2a)$ and $(3a, 5a)$, where $a \neq 0$. What is its slope?

   $2/3$

   $1/2$

   $3/2$

   $5a/3a$

2. A line is parallel to $3x - 2y = 7$ and passes through the point $(-2, 5)$. What is its equation in slope-intercept form?

   $y = -3/2 x + 2$

   $y = 3/2 x + 8$

   $y = 2/3 x + 19/3$

   $y = 3/2 x + 5$

3. Find the y-intercept of a line that is perpendicular to $y = -2/3 x + 4$ and passes through the point $(4, -1)$.

   $11/3$

   $-7/2$

   $7$

   $-7$

4. A line makes an angle of $135^\circ$ with the positive x-axis. If it passes through the point $(2, -3)$, what is its equation?

   $y = -x - 1$

   $y = -x - 3$

   $y = x - 5$

   $y = -x + 1$

5. A linear model for the cost $C$ (in dollars) of producing $x$ items is given by $C = 500 + 15x$. What does the slope of this line represent?

   The total cost of producing $15$ items.

   The fixed cost of production.

   The variable cost per item produced.

   The number of items produced.

6. What are the x and y intercepts of the line $5x - 3y = 15$?

   x-intercept is $(3,0)$, y-intercept is $(0,-5)$

   x-intercept is $(3,0)$, y-intercept is $(0,5)$

   x-intercept is $(-3,0)$, y-intercept is $(0,-5)$

   x-intercept is $(5,0)$, y-intercept is $(0,-3)$

7. Find the equation of the line passing through $(1/2, 3)$ and $(2, 1)$.

   $y = -4/3 x + 11/3$

   $y = -4/3 x + 19/3$

   $y = 4/3 x + 7/3$

   $y = -4/3 x + 3$

8. A line has the same x-intercept as $2x - 5y = 10$ and the same slope as the line passing through $(1, 4)$ and $(3, 8)$. What is its equation?

   $y = 2x + 10$

   $y = 2x - 10$

   $y = 1/2 x - 5/2$

   $y = -2x + 10$

9. The temperature $T$ in a greenhouse drops at a constant rate. At 9 AM, the temperature is $25^\circ C$. At 1 PM, it is $17^\circ C$. If $t$ is the number of hours after 9 AM, what was the temperature at 6 AM?

   $30^\circ C$

   $29^\circ C$

   $33^\circ C$

   $31^\circ C$

10. What is the angle (to the nearest degree) that the line $4x - 3y = 12$ makes with the positive x-axis?

    $53^\circ$

    $127^\circ$

    $37^\circ$

    $143^\circ$

11. A line passes through the point $(6, 0)$ and has a y-intercept of $(0, -4)$. What is the area of the triangle formed by this line and the coordinate axes?

    $24$ square units

    $12$ square units

    $10$ square units

    $18$ square units

12. A line has a slope of $3/4$ and passes through the points $(k, 5)$ and $(4, 11)$. What is the value of $k$?

    $6$

    $8$

    $-2$

    $-4$

13. Which of the following equations represents a line that passes through the second and fourth quadrants ONLY?

    $y = -3x + 2$

    $y = -2$

    $y = -x$

    $y = 2x + 1$

14. A car's value $V$ (in dollars) decreases linearly with its age $t$ (in years). When new, its value is $\$25,000$. After 5 years, its value is $\$10,000$. What will be its value when it is 8 years old?

    $\$1,500$

    $\$1,000$

    $\$2,500$

    $\$3,000$

15. A line has an x-intercept of $a$ and a y-intercept of $b$, where $a \neq 0$ and $b \neq 0$. What is the slope of this line in terms of $a$ and $b$?

    $a/b$

    $b/a$

    $-(b/a)$

    $-(a/b)$

16. At what point do the lines $2x + y = 7$ and $3x - 2y = 0$ intersect?

    $(1, 5)$

    $(3, 2)$

    $(4, -1)$

    $(2, 3)$

17. A line has a slope of $1/2$ and passes through the midpoint of the line segment connecting $(1, -2)$ and $(5, 6)$. What is its equation?

    $y = 1/2 x - 1$

    $y = 1/2 x + 1/2$

    $y = 1/2 x + 2$

    $y = 1/2 x + 1$

18. A graph shows the amount of fuel remaining in a tank (liters) versus distance traveled (km). The line passes through $(0, 60)$ and $(300, 15)$. What does the y-intercept represent in this context?

    The distance traveled on a full tank.

    The distance traveled before the tank is empty.

    The initial amount of fuel in the tank.

    The rate of fuel consumption per km.

19. The vertices of a triangle are $A(0,0)$, $B(4,0)$, and $C(2,6)$. What is the equation of the line passing through point $C$ and parallel to side $AB$?

    $x = 6$

    $y = 6$

    $x = 2$

    $y = 2$

20. Convert the equation $2x - 5y + 15 = 0$ into slope-intercept form ($y = mx + b$).

    $y = 2/5 x + 3$

    $y = -2/5 x + 3$

    $y = 2/5 x - 3$

    $y = 5/2 x - 15/2$

21. A line passes through the points $(k, 5)$ and $(1, 3)$. If the angle of inclination of this line with the positive x-axis is $135^\circ$, what is the value of $k$?

    $3$

    $-3$

    $-1$

    $1$

22. A line $L_1$ has the equation $3x - 4y = 12$. Line $L_2$ is perpendicular to $L_1$ and passes through the point $(-3, 5)$. What is the equation of line $L_2$?

    $4x + 3y = 3$

    $4x - 3y = -27$

    $4x + 3y = 9$

    $3x + 4y = 11$

23. The cost $C$ (in dollars) of producing $n$ units of a certain product is given by the linear equation $C = 15n + 250$. What does the $C$-intercept represent in this context?

    The cost of producing one unit.

    The total revenue from producing $250$ units.

    The fixed cost, regardless of the number of units produced.

    The number of units that can be produced for free.

24. The points $(1, k)$, $(3, 11)$, and $(5, 17)$ are collinear. Find the value of $k$.

    $2$

    $4$

    $7$

    $5$

25. A line has the equation $5x - 2y = 20$. Find the area of the triangle formed by this line and the coordinate axes.

    $50$ square units

    $10$ square units

    $20$ square units

    $40$ square units

26. Which of the following statements is true about the line represented by the equation $y - 4 = -(3/5)(x + 2)$?

    The x-intercept is $14/5$.

    The y-intercept is $2$.

    The y-intercept is $14/5$.

    The slope is $3/5$.

27. A car's value depreciates linearly. In $2018$, its value was $\$25,000$. In $2022$, its value was $\$17,000$. Assuming the depreciation continues at the same rate, what will be the car's value in $2025$?

    $\$15,000$

    $\$11,000$

    $\$13,000$

    $\$9,000$

28. If the line $ax + 2y = 12$ has an angle of inclination of $120^\circ$, what is the value of $a$?

    $2\sqrt{3}$

    $-\sqrt{3}/2$

    $\sqrt{3}/2$

    $-2\sqrt{3}$

29. A line has a positive slope and a negative y-intercept. Which quadrants does this line \emph{not} pass through?

    Quadrant III

    Quadrant IV

    Quadrant I

    Quadrant II

30. Line $L_1$ has equation $y = 2x + 3$. Line $L_2$ passes through $(1, 7)$ and is parallel to $L_1$. Line $L_3$ passes through $(4, 1)$ and is perpendicular to $L_2$. What is the x-intercept of $L_3$?

    $2$

    $1$

    $6$

    $4$

31. The lines $2x + y = 7$ and $3x - 2y = 0$ intersect at a point $(a, b)$. What is the equation of a line with slope $5$ that passes through $(a, b)$?

    $y = 5x + 7$

    $y = -5x + 13$

    $y = 5x - 3$

    $y = 5x - 7$

32. A linear model for the height $H$ (in cm) of a plant $t$ weeks after planting is given by $H = 1.5t + 12$. Which of the following statements correctly interprets the slope of this model?

    The plant grows $1.5$ cm per week.

    The plant grows $12$ cm every $1.5$ weeks.

    The plant's height after $1.5$ weeks is $12$ cm.

    The plant was $12$ cm tall when planted.

33. Find the equation of the perpendicular bisector of the line segment connecting points $A(2, 5)$ and $B(8, -1)$.

    $y = x - 3$

    $y = x + 3$

    $y = -x - 3$

    $y = -x + 7$

34. A line passes through the point $(0, -5)$ and has a slope of $4/3$. What is the distance from the origin $(0, 0)$ to this line?

    $2.5$

    $3$

    $4$

    $5$

35. The line passing through $(a, 3)$ and $(7, 5)$ is parallel to the line passing through $(1, 8)$ and $(3, 12)$. Find the value of $a$.

    $6$

    $5$

    $4$

    $8$

36. A line forms a triangle of area $18$ square units with the coordinate axes in the first quadrant. If the slope of the line is $-1$, what are its x- and y-intercepts?

    x-intercept is $(9,0)$, y-intercept is $(0,4)$

    x-intercept is $(3,0)$, y-intercept is $(0,6)$

    x-intercept is $(6,0)$, y-intercept is $(0,6)$

    x-intercept is $(12,0)$, y-intercept is $(0,3)$

37. The number of active users $N$ on a social media platform decreases linearly over time $t$ (in months). Initially, there were $200,000$ users. After $5$ months, the number of users was $185,000$. What is the percentage decrease in the number of users per month relative to the initial number?

    $0.75\%$

    $2.5\%$

    $1.0\%$

    $1.5\%$

38. A line passes through the point $(5, -4)$ and has an x-intercept of $3$. Find the equation of this line.

    $y = -2x + 6$

    $y = -2x - 14$

    $y = -2x + 10$

    $y = 2x - 6$

39. Line $A$ has equation $y = -(1/2)x + 3$. Line $B$ passes through $(1, 4)$ and $(3, 2)$. Which statement is true when comparing the two lines?

    Line A has a smaller slope and a smaller y-intercept than Line B.

    Line A has a greater slope and a smaller y-intercept than Line B.

    Line A has a greater slope and a greater y-intercept than Line B.

    Line A has a smaller slope and a greater y-intercept than Line B.

40. A line has a slope of $3/4$ and its y-intercept is $-(9/2)$. If the point $(k, 0)$ lies on this line, what is the value of $k$?

    $-(3/2)$

    $6$

    $-(9/4)$

    $3$