Multiple Choice

1. A linear equation is given by $3x - 2y = 8$. When sketching this graph using the gradient-intercept method, what are the values of the gradient ($m$) and the y-intercept ($c$)?

   $m = -3/2$, $c = 4$

   $m = -2$, $c = 3$

   $m = 3/2$, $c = -4$

   $m = 3$, $c = -8$

2. If a linear graph defined by $y = mx + c$ has a negative gradient ($m < 0$) and a positive y-intercept ($c > 0$), which set of quadrants must the line pass through? (Assume standard Quadrant numbering I, II, III, IV).

   Quadrants I and II only

   Quadrants I, II, and III

   Quadrants II, III, and IV

   Quadrants I, II, and IV

3. Find the x-intercept of the line given by the equation $4y - x = 12$.

   $(-3, 0)$

   $(12, 0)$

   $(0, 3)$

   $(-12, 0)$

4. A student sketches the line $y = (1/3)x - 5$. If the student keeps the gradient constant but increases the y-intercept ($c$) by 7 units, how does the new line relate to the original sketch?

   The line shifts vertically downwards by 7 units, maintaining the same slope.

   The line shifts vertically upwards by 7 units, maintaining the same slope.

   The line pivots, becoming steeper and crossing the y-axis at $(0, 7)$.

   The line shifts horizontally 7 units to the left.

5. The line is defined by the equation $y = (-2/3)x - 1$. Which of the following coordinates does *not* lie on the graph?

   $(2, -5)$

   $(0, -1)$

   $(-3, 1)$

   $(3, -3)$

6. Given two lines, $L_1: y = 3x - 4$ and $L_2: y = kx + 8$. If $L_2$ has the same x-intercept as $L_1$, what must be the value of $k$?

   $k = -2$

   $k = -6$

   $k = 6$

   $k = 3$

7. What are the x and y intercepts, respectively, for the line defined by the equation $4x - 3y = 10$?

   $(\frac{2}{5}, 0)$ and $(0, -\frac{3}{10})$.

   $(\frac{5}{2}, 0)$ and $(0, -\frac{10}{3})$.

   $(2.5, 0)$ and $(0, -3)$.

   $(-2.5, 0)$ and $(0, 3.33)$.

8. A line is defined by the equation $5(y - 1) = 2x + 15$. Find the sum of the x-intercept and the y-intercept.

   $14$.

   $-14$.

   $6$.

   $-6$.

9. The line $\frac{x}{a} + \frac{y}{b} = 1$ is an intercept form where $a$ is the x-intercept and $b$ is the y-intercept. If the equation is $2x + 7y = -14$, what are the values of $a$ and $b$?

   $a=-7, b=-2$.

   $a=-2, b=-7$.

   $a=-14, b=-14$.

   $a=7, b=2$.

10. Which statement accurately describes the intercepts of the graph defined by the equation $3x + 12 = 0$?

    The x-intercept is $(0, -4)$ and the y-intercept is $(0, 0)$.

    The line passes through the origin, so both intercepts are $(0, 0)$.

    The x-intercept is $(-4, 0)$, and there is no y-intercept.

    The x-intercept is $(4, 0)$ and the y-intercept is $(0, -12)$.

11. A line has an x-intercept of $-3$ and a y-intercept of $\frac{1}{2}$. Which equation represents this line?

    $3x + 2y = 6$.

    $x - 6y = -3$.

    $x + 6y = -3$.

    $x - 3y = 3$.

12. A linear graph passes through Quadrants I, II, and IV only. Based on this information, what must be true about the intercepts?

    Both the x-intercept and the y-intercept must be positive.

    The slope is positive, meaning the intercepts must have opposite signs.

    Both the x-intercept and the y-intercept must be negative.

    The x-intercept must be positive, and the y-intercept must be negative.

13. The linear equation $2x + 5y = C$ has an x-intercept at $(15, 0)$. What is the y-intercept of this line?

    $(0, 15)$.

    $(0, 5)$.

    $(0, 3)$.

    $(0, 6)$.

14. Given the equation $y = \frac{2}{3}x - 5$, calculate the distance between the x-intercept and the y-intercept.

    $\frac{10\sqrt{3}}{3}$.

    $\frac{\sqrt{175}}{4}$.

    $\sqrt{30}$.

    $\frac{5\sqrt{13}}{2}$.

15. A line is defined by the equation $4x + 3y = 24$. Calculate the distance, in units, between the x-intercept and the y-intercept of this line.

    $2\sqrt{5}$

    $10$

    $24$

    $14$

16. Determine the exact distance between the intercepts for the line given by the equation $y = -\frac{1}{2}x + 4$.

    $2\sqrt{10}$

    $4\sqrt{2}$

    $4\sqrt{5}$

    $12$

17. The equation of a line is given as $y = \frac{5}{6}x + \frac{5}{2}$. Find the exact distance between the points where the line crosses the coordinate axes.

    $\frac{25}{4}$

    $5.5$

    $\frac{\sqrt{61}}{2}$

    $\frac{\sqrt{50}}{2}$

18. Write the equation of the line passing through $(6, -5)$ with a slope of $\frac{2}{3}$, expressed in slope-intercept form.

    $y = \frac{2}{3}x - 1$

    $y = \frac{2}{3}x - 13$

    $y = \frac{2}{3}x - 9$

    $y = \frac{2}{3}x + 9$

19. A line has a slope of $-\frac{1}{4}$ and passes through the point $(-8, 1)$. Write its equation in standard form ($Ax+By=C$, where $A, B, C$ are integers and $A>0$).

    $4x + y = -4$

    $x + 4y = -4$

    $x + 4y = 12$

    $x - 4y = -4$

20. Find the equation of the line with slope $m=3$ passing through the point $(0.5, -2)$. Express the constant $b$ as an improper fraction in its final slope-intercept form.

    $y = 3x + \frac{7}{2}$

    $y = 3x - 2.5$

    $y = 3x - \frac{7}{2}$

    $y = 3x - \frac{1}{2}$

21. Determine the slope-intercept equation of the line that has a slope of $-1$ and passes through the point $(10, -7)$.

    $y = x - 17$

    $y = -x - 3$

    $y = -x + 3$

    $y = -x + 17$

22. The equation of a line is written as $y + 9 = -\frac{5}{2}(x - 4)$. Which statement correctly describes the line?

    The line has a slope of $-\frac{2}{5}$ and passes through $(4, -9)$.

    The line has a slope of $-\frac{5}{2}$ and passes through $(-9, 4)$.

    The line has a slope of $-\frac{5}{2}$ and passes through $(4, -9)$.

    The line has a slope of $\frac{5}{2}$ and passes through $(-4, 9)$.

23. Write the equation $y+1 = \frac{5}{6}(x+3)$ in standard form ($Ax+By=C$, where $A, B, C$ are integers and $A>0$).

    $5x - 6y = -9$

    $5x + 6y = -9$

    $6x - 5y = 9$

    $5x - 6y = 9$

24. Write the equation of the line that has a slope of zero and passes through the point $(42, -11)$.

    $x = 42$

    $y = 42x - 11$

    $y = -11x + 42$

    $y = -11$

25. Find the slope-intercept equation for the line passing through $(-4, 1)$ with a slope of $m = -1.5$.

    $y = -1.5x - 7$

    $y = -1.5x + 5$

    $y = -1.5x - 5$

    $y = 1.5x - 5$