Question

write an equation of the line that passes through each pair of points:11. (0, -4), (5, -4)12. (-2, -2), (4, 0)13. (-2, -3), (4, 5)14. 15. (-3, 0), (1, -6)16. (1, 0), (5, -1)17. (8, 2), (-2, 6)18. (-6, 5), (-6, -4)19. (8, -2), (7, -1)20. (5, -3), (2, 5)21. $left(\frac{7}{4}, 1
ight), left(-\frac{1}{4}, \frac{3}{4}
ight)$22. $left(\frac{4}{11}, -1
ight), left(-\frac{2}{3}, \frac{1}{3}
ight)$example 423. guitar lydia wants to purchase guitar lessons. she sees a sign that gives the prices for 7 guitar lessons and 11 guitar lessons. write a linear equation to find the total cost c for g lessons.guitar lessons7 lessons = $8211 lessons = $12224. census the population of laredo, texas, was about 215,500 in 2007. it was about 123,000 in 1990. if we assume that the population growth is constant, write a linear equation with an integer slope to represent p, laredos population t years after 1990.

Explanation:

Problem 11: Points $(0,-4),(5,-4)$

Step1: Calculate slope $m$

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
$m=\frac{-4-(-4)}{5-0}=\frac{0}{5}=0$

Step2: Find y-intercept $b$

Use $(0,-4)$: $b=-4$

Step3: Write line equation

$y=mx+b$
$y=0x-4$

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Problem 12: Points $(-2,-2),(4,0)$

Step1: Calculate slope $m$

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

Step2: Find y-intercept $b$

Use $(4,0)$: $0=\frac{1}{3}(4)+b$
$b=-\frac{4}{3}$

Step3: Write line equation

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

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Problem 13: Points $(-2,-3),(4,5)$

Step1: Calculate slope $m$

$m=\frac{5-(-3)}{4-(-2)}=\frac{8}{6}=\frac{4}{3}$

Step2: Find y-intercept $b$

Use $(-2,-3)$: $-3=\frac{4}{3}(-2)+b$
$-3=-\frac{8}{3}+b$
$b=-3+\frac{8}{3}=-\frac{1}{3}$

Step3: Write line equation

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

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Problem 15: Points $(-3,0),(1,-6)$

Step1: Calculate slope $m$

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

Step2: Find y-intercept $b$

Use $(-3,0)$: $0=-\frac{3}{2}(-3)+b$
$0=\frac{9}{2}+b$
$b=-\frac{9}{2}$

Step3: Write line equation

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

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Problem 16: Points $(1,0),(5,-1)$

Step1: Calculate slope $m$

$m=\frac{-1-0}{5-1}=-\frac{1}{4}$

Step2: Find y-intercept $b$

Use $(1,0)$: $0=-\frac{1}{4}(1)+b$
$b=\frac{1}{4}$

Step3: Write line equation

$y=-\frac{1}{4}x+\frac{1}{4}$

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Problem 18: Points $(-6,5),(-6,-4)$

Step1: Calculate slope $m$

$m=\frac{-4-5}{-6-(-6)}=\frac{-9}{0}$ (undefined)

Step2: Write vertical line equation

Vertical line: $x=-6$

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Problem 19: Points $(8,-2),(7,-1)$

Step1: Calculate slope $m$

$m=\frac{-1-(-2)}{7-8}=\frac{1}{-1}=-1$

Step2: Find y-intercept $b$

Use $(8,-2)$: $-2=-1(8)+b$
$-2=-8+b$
$b=6$

Step3: Write line equation

$y=-x+6$

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Problem 20: Points $(5,-3),(2,5)$

Step1: Calculate slope $m$

$m=\frac{5-(-3)}{2-5}=\frac{8}{-3}=-\frac{8}{3}$

Step2: Find y-intercept $b$

Use $(5,-3)$: $-3=-\frac{8}{3}(5)+b$
$-3=-\frac{40}{3}+b$
$b=-3+\frac{40}{3}=\frac{31}{3}$

Step3: Write line equation

$y=-\frac{8}{3}x+\frac{31}{3}$

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Problem 23: Guitar Lessons

Step1: Define variables

Let $x$ = number of lessons, $C$ = total cost. Points: $(7,82),(11,122)$

Step2: Calculate slope (cost per lesson)

$m=\frac{122-82}{11-7}=\frac{40}{4}=10$

Step3: Find fixed cost $b$

Use $(7,82)$: $82=10(7)+b$
$82=70+b$
$b=12$

Step4: Write cost equation

$C=10x+12$

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Problem 24: Laredo Population

Step1: Define variables

Let $t$ = years after 1990, $p$ = population. Points: $(0,123000),(17,215500)$

Step2: Calculate slope $m$

$m=\frac{215500-123000}{17-0}=\frac{92500}{17}=5441$ (rounded to integer)

Step3: Find initial population $b$

$b=123000$ (1990 population)

Step4: Write population equation

$p=5441t+123000$

Answer:

1. $y=-4$
2. $y=\frac{1}{3}x-\frac{4}{3}$
3. $y=\frac{4}{3}x-\frac{1}{3}$
4. $y=-\frac{3}{2}x-\frac{9}{2}$
5. $y=-\frac{1}{4}x+\frac{1}{4}$
6. $x=-6$
7. $y=-x+6$
8. $y=-\frac{8}{3}x+\frac{31}{3}$
9. $C=10x+12$
10. $p=5441t+123000$

(Note: Problems 14,17,21,22 are illegible and could not be solved from the image.)