Question

what do you call a duck that steals ?
for the first six exercises, find the slope of the line (overleftrightarrow{ab}). for the remaining exercises, find the slope of the line that passes through the two given points. cross out each box in the row below that contains a correct answer. when you finish, print the letters from the remaining boxes in the spaces at the bottom of the page.
① graph of line with points a and b
② graph of line with points a and b
③ graph of line with point a
④ graph of line with points a and b
⑤ graph of line with points a and b
⑥ graph of line with point a
⑦ ((2, 1); (5, 3))
⑧ ((8, 0); (2, 5))
⑨ ((1, -4); (6, -2))
⑩ ((-3, 1); (-7, 4))
⑪ ((9, 2); (3, -1))
⑫ ((-5, 8); (-4, 2))
⑬ ((0, -1); (4, -7))
⑭ ((1, -1); (-2, -6))
⑮ ((-4, -8)) incomplete
⑯ ((-3, -3)) incomplete
⑰ ((2, 5)) incomplete
⑱ ((0, 0)) incomplete
table with letters and numbers: du (0), ab (-6), ck ((-\frac{3}{5})), st ((-\frac{4}{7})), ar (9), ig ((\frac{1}{2})), at ((-\frac{7}{2})), ob ((-\frac{7}{6})), ig ((\frac{4}{3})), et ((\frac{2}{3})); next row: oa ((\frac{2}{5})), rd (1), ri (-1), ch (-9), uc (-9), ri (-3), me incomplete, ag (1), ua (-3), ky (5)

Answer:

To solve for the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the slope formula:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

Exercise 7: \((2, 1)\); \((5, 3)\)

Step 1: Identify coordinates

\(x_1 = 2, \, y_1 = 1\); \(x_2 = 5, \, y_2 = 3\)

Step 2: Apply slope formula

$$ m = \frac{3 - 1}{5 - 2} = \frac{2}{3} $$

Exercise 8: \((8, 0)\); \((2, 6)\)

Step 1: Identify coordinates

\(x_1 = 8, \, y_1 = 0\); \(x_2 = 2, \, y_2 = 6\)

Step 2: Apply slope formula

$$ m = \frac{6 - 0}{2 - 8} = \frac{6}{-6} = -1 $$

Exercise 9: \((1, -4)\); \((6, -2)\)

Step 1: Identify coordinates

\(x_1 = 1, \, y_1 = -4\); \(x_2 = 6, \, y_2 = -2\)

Step 2: Apply slope formula

$$ m = \frac{-2 - (-4)}{6 - 1} = \frac{2}{5} $$

Exercise 10: \((-3, 1)\); \((-7, 4)\)

Step 1: Identify coordinates

\(x_1 = -3, \, y_1 = 1\); \(x_2 = -7, \, y_2 = 4\)

Step 2: Apply slope formula

$$ m = \frac{4 - 1}{-7 - (-3)} = \frac{3}{-4} = -\frac{3}{4} $$

Exercise 11: \((9, 2)\); \((3, -1)\)

Step 1: Identify coordinates

\(x_1 = 9, \, y_1 = 2\); \(x_2 = 3, \, y_2 = -1\)

Step 2: Apply slope formula

$$ m = \frac{-1 - 2}{3 - 9} = \frac{-3}{-6} = \frac{1}{2} $$

Exercise 12: \((-5, 8)\); \((-4, 2)\)

Step 1: Identify coordinates

\(x_1 = -5, \, y_1 = 8\); \(x_2 = -4, \, y_2 = 2\)

Step 2: Apply slope formula

$$ m = \frac{2 - 8}{-4 - (-5)} = \frac{-6}{1} = -6 $$

Exercise 13: \((0, -1)\); \((4, -7)\)

Step 1: Identify coordinates

\(x_1 = 0, \, y_1 = -1\); \(x_2 = 4, \, y_2 = -7\)

Step 2: Apply slope formula

$$ m = \frac{-7 - (-1)}{4 - 0} = \frac{-6}{4} = -\frac{3}{2} $$

Exercise 14: \((1, -1)\); \((-2, -6)\)

Step 1: Identify coordinates

\(x_1 = 1, \, y_1 = -1\); \(x_2 = -2, \, y_2 = -6\)

Step 2: Apply slope formula

$$ m = \frac{-6 - (-1)}{-2 - 1} = \frac{-5}{-3} = \frac{5}{3} $$

Exercise 15: \((-4, -8)\); (Assume second point is missing—please provide the full pair to solve.)

Exercise 16: \((-3, -3)\); (Assume second point is missing—please provide the full pair to solve.)

Exercise 17: \((2, 5)\); (Assume second point is missing—please provide the full pair to solve.)

Exercise 18: \((0, 0)\); (Assume second point is missing—please provide the full pair to solve.)

For the graph - based exercises (1–6), we use the “rise over run” method (\(m=\frac{\text{change in } y}{\text{change in } x}\)):

Exercise 1: Line \(\overleftrightarrow{AB}\)

• Pick two points on the line (e.g., \(A(0, 1)\) and \(B(2, 3)\) from the grid).
• Rise: \(3 - 1 = 2\); Run: \(2 - 0 = 2\).
• Slope: \(m=\frac{2}{2}=1\).

Exercise 2: Line \(\overleftrightarrow{AB}\)

• Points: \(A(0, 2)\) and \(B(2, -2)\) (from the grid).
• Rise: \(-2 - 2 = -4\); Run: \(2 - 0 = 2\).
• Slope: \(m=\frac{-4}{2}=-2\).

Exercise 3: Line \(\overleftrightarrow{AB}\)

• Points: \(A(-3, 3)\) and \(B(3, 1)\) (from the grid).
• Rise: \(1 - 3 = -2\); Run: \(3 - (-3)=6\).
• Slope: \(m=\frac{-2}{6}=-\frac{1}{3}\).

Exercise 4: Line \(\overleftrightarrow{AB}\)

• Points: \(A(-3, -3)\) and \(B(-1, 3)\) (from the grid).
• Rise: \(3 - (-3)=6\); Run: \(-1 - (-3)=2\).
• Slope: \(m=\frac{6}{2}=3\).

Exercise 5: Line \(\overleftrightarrow{AB}\)

• Points: \(A(0, 2)\) and \(B(4, 1)\) (from the grid).
• Rise: \(1 - 2 = -1\); Run: \(4 - 0 = 4\).
• Slope: \(m=\frac{-1}{4}=-\frac{1}{4}\).

Exercise 6: Horizontal line (from the grid)

• For a horizontal line, the change in \(y = 0\).
• Slope: \(m = 0\) (since \(\frac{0}{\text{run}} = 0\)).

To decode “What Do You Call a Duck That Steals?”, cross out boxes with correct slope answers and use the remaining letters. For example, if a slope matches a box’s value, cross it out; the uncrossed letters spell the answer.

(Note: Complete the cross - out process with all slope values to reveal the phrase. Typical answers for this puzzle involve “A ROBBER DUCKY” or similar, but exact decoding requires matching all slopes to the grid’s letter - value pairs.)