Question

writing equations of linear functions
use the given information in each problem to write an equation in slope - intercept form. use the value of b, the y - intercept, and its corresponding letter to solve the riddle at the bottom.
1
slope = $-\frac{1}{6}$; (12, -2)
y - intercept:
equation:
2
slope = 2; (14, 33)
y - intercept:
equation:
3
slope = -7; (4, -30)
y - intercept:
equation:
4
slope = $\frac{1}{2}$; (-2, 7)
y - intercept:
equation:
5
(-3, -14) and (0, -9)
y - intercept:
equation:
6
(-2, 5) and (2, 1)
y - intercept:
equation:
7
(7, -1) and (21, -5)
y - intercept:
equation:
8
(1, 9) and (6, 34)
y - intercept:
equation:
p: -9
a: -8
h: 3
s: 1
l: 8
e: 4
o: 5
c: 9
k: -4
d: 7
t: -2
i: 0
what did the mathematician do to practice over winter break?
$\overline{6}$ $\overline{1}$ $\overline{3}$ $\overline{3}$ $\overline{6}$ $\overline{8}$ $\overline{7}$ $\overline{4}$ $\overline{2}$ $\overline{5}$ $\overline{8}$ $\overline{7}$

Explanation:

Step1: Recall slope-intercept form

Slope-intercept form is $y = mx + b$, where $m$ = slope, $b$ = y-intercept. Substitute given $m, x, y$ to solve for $b$, then write the equation.

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Problem 1

Step1: Substitute values into formula

$m = -\frac{1}{6}$, $x=12$, $y=-2$:
$-2 = -\frac{1}{6}(12) + b$

Step2: Simplify and solve for $b$

$-2 = -2 + b \implies b = 0$

Step3: Write the final equation

$y = -\frac{1}{6}x + 0$

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Problem 2

Step1: Substitute values into formula

$m = 2$, $x=14$, $y=33$:
$33 = 2(14) + b$

Step2: Simplify and solve for $b$

$33 = 28 + b \implies b = 5$

Step3: Write the final equation

$y = 2x + 5$

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Problem 3

Step1: Substitute values into formula

$m = -7$, $x=4$, $y=-30$:
$-30 = -7(4) + b$

Step2: Simplify and solve for $b$

$-30 = -28 + b \implies b = -2$

Step3: Write the final equation

$y = -7x - 2$

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Problem 4

Step1: Substitute values into formula

$m = \frac{1}{2}$, $x=-2$, $y=7$:
$7 = \frac{1}{2}(-2) + b$

Step2: Simplify and solve for $b$

$7 = -1 + b \implies b = 8$

Step3: Write the final equation

$y = \frac{1}{2}x + 8$

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Problem 5

Step1: Calculate slope first

Points $(-3,-14)$ and $(0,-9)$:
$m = \frac{-9 - (-14)}{0 - (-3)} = \frac{5}{3}$

Step2: Identify y-intercept

The point $(0,-9)$ gives $b = -9$

Step3: Write the final equation

$y = \frac{5}{3}x - 9$

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Problem 6

Step1: Calculate slope first

Points $(-2,5)$ and $(2,1)$:
$m = \frac{1 - 5}{2 - (-2)} = \frac{-4}{4} = -1$

Step2: Substitute to find $b$

Use $(2,1)$: $1 = -1(2) + b \implies b = 3$

Step3: Write the final equation

$y = -x + 3$

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Problem 7

Step1: Calculate slope first

Points $(7,-1)$ and $(21,-5)$:
$m = \frac{-5 - (-1)}{21 - 7} = \frac{-4}{14} = -\frac{2}{7}$

Step2: Substitute to find $b$

Use $(7,-1)$: $-1 = -\frac{2}{7}(7) + b \implies -1 = -2 + b \implies b = 1$

Step3: Write the final equation

$y = -\frac{2}{7}x + 1$

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Problem 8

Step1: Calculate slope first

Points $(1,9)$ and $(6,34)$:
$m = \frac{34 - 9}{6 - 1} = \frac{25}{5} = 5$

Step2: Substitute to find $b$

Use $(1,9)$: $9 = 5(1) + b \implies b = 4$

Step3: Write the final equation

$y = 5x + 4$

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Riddle Solution

Step1: Match $b$ values to letters

$b$ values (1-8): $0(I), 5(O), -2(T), 8(L), -9(P), 3(H), 1(S), 4(E)$

Step2: Map to riddle blanks

Blanks: $\frac{6}{b_6=3(H)}, \frac{1}{b_1=0(I)}, \frac{3}{b_3=-2(T)}, \frac{3}{b_6=3(H)}, \frac{6}{b_6=3(H)}, \frac{8}{b_4=8(L)}, \frac{7}{b_7=1(S)}, \frac{4}{b_8=4(E)}, \frac{2}{b_2=5(O)}, \frac{5}{b_5=-9(P)}, \frac{8}{b_4=8(L)}, \frac{7}{b_7=1(S)}$

Answer:

Individual Problem Results:

1. y-intercept: $0$, equation: $y = -\frac{1}{6}x$
2. y-intercept: $5$, equation: $y = 2x + 5$
3. y-intercept: $-2$, equation: $y = -7x - 2$
4. y-intercept: $8$, equation: $y = \frac{1}{2}x + 8$
5. y-intercept: $-9$, equation: $y = \frac{5}{3}x - 9$
6. y-intercept: $3$, equation: $y = -x + 3$
7. y-intercept: $1$, equation: $y = -\frac{2}{7}x + 1$
8. y-intercept: $4$, equation: $y = 5x + 4$

Riddle Answer:

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