Question

write the standard form of the equation of the line that is perpendicular to the graph of the given equation and that passes through the point with the given coordinates.

1. $2x - y + 6 = 0; (0, -3)$ 11. $2x - 5y - 6 = 0; (-4, 2)$ 12. $3x + 4y - 13 = 0; (2, 7)$

1. consumerism marillia paid $180 for 3 video games and 4 books. three months later she purchased 8 books and 6 video games. her brother guessed that she spent $320. assuming that the prices of video games and books did not change, is it possible that she spent $320 for the second set of purchases? explain.

Explanation:

For Questions 10, 11, 12:

Question 10

Step1: Find original slope

Rewrite $2x - y + 6 = 0$ as $y=2x+6$, so slope $m_1=2$.

Step2: Find perpendicular slope

Perpendicular slope $m_2=-\frac{1}{m_1}=-\frac{1}{2}$.

Step3: Use point-slope form

Point $(0,-3)$: $y - (-3)=-\frac{1}{2}(x-0)$

Step4: Convert to standard form

$y+3=-\frac{1}{2}x$ → $x + 2y + 6 = 0$

Question 11

Step1: Find original slope

Rewrite $2x - 5y - 6 = 0$ as $y=\frac{2}{5}x-\frac{6}{5}$, so slope $m_1=\frac{2}{5}$.

Step2: Find perpendicular slope

Perpendicular slope $m_2=-\frac{5}{2}$.

Step3: Use point-slope form

Point $(-4,2)$: $y - 2=-\frac{5}{2}(x+4)$

Step4: Convert to standard form

$2(y-2)=-5(x+4)$ → $5x + 2y + 16 = 0$

Question 12

Step1: Find original slope

Rewrite $3x + 4y - 13 = 0$ as $y=-\frac{3}{4}x+\frac{13}{4}$, so slope $m_1=-\frac{3}{4}$.

Step2: Find perpendicular slope

Perpendicular slope $m_2=\frac{4}{3}$.

Step3: Use point-slope form

Point $(2,7)$: $y - 7=\frac{4}{3}(x-2)$

Step4: Convert to standard form

$3(y-7)=4(x-2)$ → $4x - 3y + 13 = 0$

Step1: Define variables

Let $v$ = cost of 1 video game, $b$ = cost of 1 book.

Step2: Set up first equation

$3v + 4b = 180$

Step3: Analyze second purchase

Second purchase: $6v + 8b = 2(3v + 4b)$

Step4: Substitute first equation

$2(180) = 360$

Step5: Compare to guessed amount

$360
eq 320$, so not possible.

Answer:

1. $x + 2y + 6 = 0$
2. $5x + 2y + 16 = 0$
3. $4x - 3y + 13 = 0$

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For Question 13: