Question

math cartoon
each pair of points. place the letter of the problem in the space above used more than once.
i) (3, 7), (3, -5)
k) $\left(\frac{1}{2}, \frac{2}{3}\
ight)$, $\left(\frac{-1}{4}, \frac{-2}{3}\
ight)$
s) $\left(\frac{3}{5}, \frac{5}{7}\
ight)$, $\left(\frac{2}{5}, \frac{-3}{7}\
ight)$
h) $\left(\frac{3}{8}, \frac{-1}{6}\
ight)$, $\left(\frac{1}{4}, \frac{3}{4}\
ight)$
d) $\left(1\frac{1}{2}, 3\frac{2}{3}\
ight)$, $\left(-1\frac{1}{2}, \frac{-1}{3}\
ight)$
f) $\left(5, 2\frac{3}{8}\
ight)$, $\left(3\frac{1}{2}, -4\frac{1}{2}\
ight)$
n) $\left(10\frac{1}{3}, -2\frac{3}{4}\
ight)$, $\left(8, 4\frac{1}{3}\
ight)$

Explanation:

Step1: Recall slope formula

The slope \(M\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M=\frac{y_2-y_1}{x_2-x_1}\)

Step2: Calculate slope for I

Points \((3,7),(3,-5)\): \(x_1=x_2=3\), denominator is 0, slope is undefined.

Step3: Calculate slope for K

Points \((\frac{1}{2},\frac{2}{3}),(-\frac{1}{4},-\frac{2}{3})\):
\(M=\frac{-\frac{2}{3}-\frac{2}{3}}{-\frac{1}{4}-\frac{1}{2}}=\frac{-\frac{4}{3}}{-\frac{3}{4}}=\frac{16}{9}\)

Step4: Calculate slope for S

Points \((\frac{3}{5},\frac{5}{7}),(\frac{2}{5},-\frac{3}{7})\):
\(M=\frac{-\frac{3}{7}-\frac{5}{7}}{\frac{2}{5}-\frac{3}{5}}=\frac{-\frac{8}{7}}{-\frac{1}{5}}=\frac{40}{7}\)

Step5: Calculate slope for H

Points \((\frac{3}{8},-\frac{1}{6}),(\frac{1}{4},\frac{3}{4})\):
\(M=\frac{\frac{3}{4}-(-\frac{1}{6})}{\frac{1}{4}-\frac{3}{8}}=\frac{\frac{11}{12}}{-\frac{1}{8}}=-\frac{22}{3}\)

Step6: Calculate slope for O

Points \((1\frac{1}{2},3\frac{2}{3}),(-1\frac{1}{2},-\frac{1}{3})\): Convert to improper fractions \((\frac{3}{2},\frac{11}{3}),(-\frac{3}{2},-\frac{1}{3})\)
\(M=\frac{-\frac{1}{3}-\frac{11}{3}}{-\frac{3}{2}-\frac{3}{2}}=\frac{-4}{-3}=\frac{4}{3}\)

Step7: Calculate slope for F

Points \((5,2\frac{3}{8}),(3\frac{1}{2},-4\frac{1}{2})\): Convert to improper fractions \((5,\frac{19}{8}),(\frac{7}{2},-\frac{9}{2})\)
\(M=\frac{-\frac{9}{2}-\frac{19}{8}}{\frac{7}{2}-5}=\frac{-\frac{55}{8}}{-\frac{3}{2}}=\frac{55}{12}\)

Step8: Calculate slope for N

Points \((10\frac{1}{3},-2\frac{3}{4}),(8,4\frac{1}{3})\): Convert to improper fractions \((\frac{31}{3},-\frac{11}{4}),(8,\frac{13}{3})\)
\(M=\frac{\frac{13}{3}-(-\frac{11}{4})}{8-\frac{31}{3}}=\frac{\frac{85}{12}}{-\frac{7}{3}}=-\frac{85}{28}\)

Answer:

I: Undefined
K: $\frac{16}{9}$
S: $\frac{40}{7}$
H: $-\frac{22}{3}$
O: $\frac{4}{3}$
F: $\frac{55}{12}$
N: $-\frac{85}{28}$