Question

write an equation of the line that passes through the given point and has slope $m$.1. $(-1, 6); m = 5$2. $(10, 3); m = -2$3. $(2, -3); m = 7$4. $(-4, -9); m = 2$5. $(5, -4); m = \frac{1}{3}$6. $(-8, 1); m = -\frac{3}{4}$write an equation of the line shown.7.8.9.write an equation of the line that passes through the given points.10. $(-10, 7), (5, -3)$11. $(-5, -3), (12, 17.4)$12. $(-8, 84), (5, -46)$write an equation for the linear function $f$ with the given values.13. $f(4) = -8, f(-3) = 1$14. $f(6) = -4, f(9) = -9$15. $f(-1) = -6, f(4) = -14$

Explanation:

Problems 1-6 (Point-Slope to Slope-Intercept)

We use the point-slope form $y - y_1 = m(x - x_1)$, then rearrange to $y=mx+b$.

Problem 1

Step1: Substitute point and slope

$y - 6 = 5(x - (-1))$

Step2: Simplify to slope-intercept

$y - 6 = 5x + 5 \implies y = 5x + 11$

Problem 2

Step1: Substitute point and slope

$y - 3 = -2(x - 10)$

Step2: Simplify to slope-intercept

$y - 3 = -2x + 20 \implies y = -2x + 23$

Problem 3

Step1: Substitute point and slope

$y - (-3) = 7(x - 2)$

Step2: Simplify to slope-intercept

$y + 3 = 7x - 14 \implies y = 7x - 17$

Problem 4

Step1: Substitute point and slope

$y - (-9) = 2(x - (-4))$

Step2: Simplify to slope-intercept

$y + 9 = 2x + 8 \implies y = 2x - 1$

Problem 5

Step1: Substitute point and slope

$y - (-4) = \frac{1}{3}(x - 5)$

Step2: Simplify to slope-intercept

$y + 4 = \frac{1}{3}x - \frac{5}{3} \implies y = \frac{1}{3}x - \frac{17}{3}$

Problem 6

Step1: Substitute point and slope

$y - 1 = -\frac{3}{4}(x - (-8))$

Step2: Simplify to slope-intercept

$y - 1 = -\frac{3}{4}x - 6 \implies y = -\frac{3}{4}x - 5$

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Problem 7

Step1: Identify slope from two points

Points $(0,5)$ and $(1,1)$: $m = \frac{1-5}{1-0} = -4$

Step2: Use y-intercept to write equation

$b=5$, so $y = -4x + 5$

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Problem 8

Step1: Calculate slope from two points

Points $(-4,-4)$ and $(-2,2)$: $m = \frac{2-(-4)}{-2-(-4)} = 3$

Step2: Find y-intercept via substitution

$2 = 3(-2) + b \implies b=8$, so $y = 3x + 8$

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Problem 9

Step1: Calculate slope from two points

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

Step2: Find y-intercept via substitution

$7 = -\frac{1}{2}(-4) + b \implies b=5$, so $y = -\frac{1}{2}x + 5$

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Problem 10

Step1: Calculate slope of two points

$m = \frac{-3-7}{5-(-10)} = \frac{-10}{15} = -\frac{2}{3}$

Step2: Find y-intercept via substitution

$7 = -\frac{2}{3}(-10) + b \implies b = \frac{1}{3}$, so $y = -\frac{2}{3}x + \frac{1}{3}$

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Problem 11

Step1: Calculate slope of two points

$m = \frac{17.4-(-3)}{12-(-5)} = \frac{20.4}{17} = 1.2 = \frac{6}{5}$

Step2: Find y-intercept via substitution

$-3 = \frac{6}{5}(-5) + b \implies b=3$, so $y = \frac{6}{5}x + 3$

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Problem 12

Step1: Calculate slope of two points

$m = \frac{-46-84}{5-(-8)} = \frac{-130}{13} = -10$

Step2: Find y-intercept via substitution

$84 = -10(-8) + b \implies b=4$, so $y = -10x + 4$

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Problem 13

Step1: Define points from function values

Points $(4,-8)$ and $(-3,1)$; $m = \frac{1-(-8)}{-3-4} = -\frac{9}{7}$

Step2: Find y-intercept via substitution

$-8 = -\frac{9}{7}(4) + b \implies b = -\frac{20}{7}$, so $y = -\frac{9}{7}x - \frac{20}{7}$

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Problem 14

Step1: Define points from function values

Points $(6,-4)$ and $(9,-9)$; $m = \frac{-9-(-4)}{9-6} = -\frac{5}{3}$

Step2: Find y-intercept via substitution

$-4 = -\frac{5}{3}(6) + b \implies b=6$, so $y = -\frac{5}{3}x + 6$

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Problem 15

Step1: Define points from function values

Points $(-1,-6)$ and $(4,-14)$; $m = \frac{-14-(-6)}{4-(-1)} = -\frac{8}{5}$

Step2: Find y-intercept via substitution

$-6 = -\frac{8}{5}(-1) + b \implies b = -\frac{38}{5}$, so $y = -\frac{8}{5}x - \frac{38}{5}$

Answer:

1. $y = 5x + 11$
2. $y = -2x + 23$
3. $y = 7x - 17$
4. $y = 2x - 1$
5. $y = \frac{1}{3}x - \frac{17}{3}$
6. $y = -\frac{3}{4}x - 5$
7. $y = -4x + 5$
8. $y = 3x + 8$
9. $y = -\frac{1}{2}x + 5$
10. $y = -\frac{2}{3}x + \frac{1}{3}$
11. $y = \frac{6}{5}x + 3$
12. $y = -10x + 4$
13. $y = -\frac{9}{7}x - \frac{20}{7}$
14. $y = -\frac{5}{3}x + 6$
15. $y = -\frac{8}{5}x - \frac{38}{5}$