Question

writing equations in slope-intercept formexample 1write an equation of the line that passes through the given point and has the given slope.1. (3, 2); slope $\frac{1}{4}$2. (3, -2); slope $\frac{1}{3}$3. (6, 4); slope $-\frac{2}{3}$4. (-5, 8); slope -15. (4, 3); slope $\frac{1}{7}$6. (1, -5); slope $-\frac{3}{4}$example 27. exercise carlos is jogging at a constant speed. he starts a timer when he is 12 feet from his starting position. after 3 seconds, carlos is 21 feet from his starting position. write a linear equation to represent the distance $d$ of carlos from his starting position $t$ seconds after starting the timer.8. jobs mr. kimball sells computer software. he earns a base salary of $41,250 and 8% commission on his sales. write an equation to represent mr. kimballs total pay $p$ after selling $d$ dollars of software.9. use a slope in 2006, the average ticket price for a national football league game was $62.38. since then, the cost has increased an average of $2.34 per year. write a linear equation to represent the cost $c$ of an nfl ticket $y$ years after 2006.10. typing nate has already typed 250 words. he then starts a timer and finds that he types 150 words in 3 minutes. if he types at a constant rate, write a linear equation to represent the number of words $w$ he types $m$ minutes after starting the timer.

Explanation:

Step1: Recall point-slope formula

The point-slope form is $y - y_1 = m(x - x_1)$, where $(x_1,y_1)$ is the point, $m$ is slope. Convert to slope-intercept $y=mx+b$.
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Problem 1: Point $(3,2)$, slope $\frac{1}{4}$

Step1: Substitute into point-slope

$y - 2 = \frac{1}{4}(x - 3)$

Step2: Rearrange to slope-intercept

$y = \frac{1}{4}x - \frac{3}{4} + 2 = \frac{1}{4}x + \frac{5}{4}$

Problem 2: Point $(3,-2)$, slope $\frac{1}{2}$

Step1: Substitute into point-slope

$y - (-2) = \frac{1}{2}(x - 3)$

Step2: Rearrange to slope-intercept

$y = \frac{1}{2}x - \frac{3}{2} - 2 = \frac{1}{2}x - \frac{7}{2}$

Problem 3: Point $(6,4)$, slope $-\frac{1}{3}$

Step1: Substitute into point-slope

$y - 4 = -\frac{1}{3}(x - 6)$

Step2: Rearrange to slope-intercept

$y = -\frac{1}{3}x + 2 + 4 = -\frac{1}{3}x + 6$

Problem 4: Point $(-5,8)$, slope $-1$

Step1: Substitute into point-slope

$y - 8 = -1(x - (-5))$

Step2: Rearrange to slope-intercept

$y = -x - 5 + 8 = -x + 3$

Problem 5: Point $(4,3)$, slope $\frac{1}{7}$

Step1: Substitute into point-slope

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

Step2: Rearrange to slope-intercept

$y = \frac{1}{7}x - \frac{4}{7} + 3 = \frac{1}{7}x + \frac{17}{7}$

Problem 6: Point $(1,-5)$, slope $-\frac{3}{4}$

Step1: Substitute into point-slope

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

Step2: Rearrange to slope-intercept

$y = -\frac{3}{4}x + \frac{3}{4} - 5 = -\frac{3}{4}x - \frac{17}{4}$
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Problem 7: Carlos's jogging distance

Step1: Define variables, find slope

Slope $m = \frac{21-12}{3-0} = 3$, initial $b=12$

Step2: Write slope-intercept equation

$d = 3t + 12$

Problem 8: Mr. Kimball's pay

Step1: Identify base pay and commission

Base pay $b=41250$, commission rate $m=0.02$

Step2: Write total pay equation

$p = 0.02d + 41250$

Problem 9: NFL ticket price

Step1: Identify initial cost and rate

2006 cost $b=62.38$, annual increase $m=2.34$

Step2: Write cost equation

$C = 2.34x + 62.38$ (where $x$ = years after 2006)

Problem 10: Nate's typing

Step1: Calculate typing rate

Rate $m = \frac{150}{3} = 50$ words/min, initial $b=250$

Step2: Write total words equation

$w = 50m + 250$ (where $m$ = minutes after timer start)

Answer:

1. $y = \frac{1}{4}x + \frac{5}{4}$
2. $y = \frac{1}{2}x - \frac{7}{2}$
3. $y = -\frac{1}{3}x + 6$
4. $y = -x + 3$
5. $y = \frac{1}{7}x + \frac{17}{7}$
6. $y = -\frac{3}{4}x - \frac{17}{4}$
7. $d = 3t + 12$
8. $p = 0.02d + 41250$
9. $C = 2.34x + 62.38$
10. $w = 50m + 250$