instructions: answer the following questions using your notes and knowledge from this unit. you must show all work on every question (including multiple choice), explain when asked to, and hand this in on time to be eligible for full credit.
due date: ______
lo1:
1. solve the following systems graphically. leave your answer as a coordinate point.
a. $y = x - 6$
$y = -\frac{1}{2}x - 3$
b. $y = -2x - 8$
$y = \frac{2}{3}x + 8$
c. $y = \frac{5}{4}x - 4$
$x + 2y = 6$
d. $5x - 2y = 10$
$3x + y = 17$
2. given the functions $f(x) = 300 - 10x$ and $g(x) = 75 + 15x$, find the value of $x$ when $f(x) = g(x)$ graphically. sketch a picture.

Question

instructions: answer the following questions using your notes and knowledge from this unit. you must show all work on every question (including multiple choice), explain when asked to, and hand this in on time to be eligible for full credit.
due date: ______
lo1:
1. solve the following systems graphically. leave your answer as a coordinate point.
   a. $y = x - 6$
      $y = -\frac{1}{2}x - 3$
   b. $y = -2x - 8$
      $y = \frac{2}{3}x + 8$
   c. $y = \frac{5}{4}x - 4$
      $x + 2y = 6$
   d. $5x - 2y = 10$
      $3x + y = 17$
2. given the functions $f(x) = 300 - 10x$ and $g(x) = 75 + 15x$, find the value of $x$ when $f(x) = g(x)$ graphically. sketch a picture.

Accepted Answer

### Part 1a: Solve $ y = x - 6 $ and $ y = -\frac{1}{2}x - 3 $ graphically
# Explanation:
## Step 1: Find intercepts for $ y = x - 6 $
- For $ x = 0 $, $ y = 0 - 6 = -6 $, so the y - intercept is $ (0, -6) $.
- For $ y = 0 $, $ 0 = x - 6 \Rightarrow x = 6 $, so the x - intercept is $ (6, 0) $.

## Step 2: Find intercepts for $ y = -\frac{1}{2}x - 3 $
- For $ x = 0 $, $ y = - 3 $, so the y - intercept is $ (0, -3) $.
- For $ y = 0 $, $ 0 = -\frac{1}{2}x - 3 \Rightarrow \frac{1}{2}x=-3\Rightarrow x = - 6 $, so the x - intercept is $ (-6, 0) $.

## Step 3: Graph the lines
- Plot the points for each line and draw the lines. The intersection point of the two lines is the solution. By solving the system algebraically (to check), set $ x - 6=-\frac{1}{2}x - 3 $.
  - Add $ \frac{1}{2}x $ to both sides: $ \frac{3}{2}x-6=-3 $.
  - Add 6 to both sides: $ \frac{3}{2}x = 3 $.
  - Multiply both sides by $ \frac{2}{3} $: $ x = 2 $.
  - Substitute $ x = 2 $ into $ y=x - 6 $, $ y=2 - 6=-4 $.

# Answer:
$(2, -4)$

### Part 1b: Solve $ y = -2x - 8 $ and $ y=\frac{2}{3}x + 8 $ graphically
# Explanation:
## Step 1: Find intercepts for $ y=-2x - 8 $
- For $ x = 0 $, $ y=-8 $, y - intercept $ (0, -8) $.
- For $ y = 0 $, $ 0=-2x - 8\Rightarrow 2x=-8\Rightarrow x=-4 $, x - intercept $ (-4, 0) $.

## Step 2: Find intercepts for $ y=\frac{2}{3}x + 8 $
- For $ x = 0 $, $ y = 8 $, y - intercept $ (0, 8) $.
- For $ y = 0 $, $ 0=\frac{2}{3}x + 8\Rightarrow\frac{2}{3}x=-8\Rightarrow x=-12 $, x - intercept $ (-12, 0) $.

## Step 3: Graph the lines and find intersection
- Solve algebraically: $ -2x - 8=\frac{2}{3}x + 8 $.
  - Multiply both sides by 3 to eliminate fraction: $ -6x-24 = 2x + 24 $.
  - Add $ 6x $ to both sides: $ -24=8x + 24 $.
  - Subtract 24 from both sides: $ -48 = 8x $.
  - Divide by 8: $ x=-6 $.
  - Substitute $ x = - 6 $ into $ y=-2x - 8 $, $ y=-2\times(-6)-8 = 12 - 8 = 4 $.

# Answer:
$(-6, 4)$

### Part 1c: Solve $ y=\frac{5}{4}x - 4 $ and $ x + 2y=6 $ (rewrite $ x + 2y=6$ as $ y=-\frac{1}{2}x + 3 $) graphically
# Explanation:
## Step 1: Find intercepts for $ y=\frac{5}{4}x - 4 $
- For $ x = 0 $, $ y=-4 $, y - intercept $ (0, -4) $.
- For $ y = 0 $, $ 0=\frac{5}{4}x - 4\Rightarrow\frac{5}{4}x = 4\Rightarrow x=\frac{16}{5}=3.2 $, x - intercept $ (3.2, 0) $.

## Step 2: Find intercepts for $ y = -\frac{1}{2}x+3 $
- For $ x = 0 $, $ y = 3 $, y - intercept $ (0, 3) $.
- For $ y = 0 $, $ 0=-\frac{1}{2}x + 3\Rightarrow\frac{1}{2}x = 3\Rightarrow x = 6 $, x - intercept $ (6, 0) $.

## Step 3: Find intersection
- Solve algebraically: $ \frac{5}{4}x - 4=-\frac{1}{2}x + 3 $.
  - Multiply by 4: $ 5x-16=-2x + 12 $.
  - Add $ 2x $ to both sides: $ 7x-16 = 12 $.
  - Add 16 to both sides: $ 7x=28 $.
  - Divide by 7: $ x = 4 $.
  - Substitute $ x = 4 $ into $ y=\frac{5}{4}x - 4 $, $ y=\frac{5}{4}\times4-4=5 - 4 = 1 $.

# Answer:
$(4, 1)$

### Part 1d: Solve $ 5x - 2y = 10 $ (rewrite as $ y=\frac{5}{2}x - 5 $) and $ 3x + y = 17 $ (rewrite as $ y=-3x + 17 $) graphically
# Explanation:
## Step 1: Find intercepts for $ y=\frac{5}{2}x - 5 $
- For $ x = 0 $, $ y=-5 $, y - intercept $ (0, -5) $.
- For $ y = 0 $, $ 0=\frac{5}{2}x - 5\Rightarrow\frac{5}{2}x = 5\Rightarrow x = 2 $, x - intercept $ (2, 0) $.

## Step 2: Find intercepts for $ y=-3x + 17 $
- For $ x = 0 $, $ y = 17 $, y - intercept $ (0, 17) $.
- For $ y = 0 $, $ 0=-3x + 17\Rightarrow3x = 17\Rightarrow x=\frac{17}{3}\approx5.67 $, x - intercept $ (\frac{17}{3}, 0) $.

## Step 3: Find intersection
- Solve algebraically: $ \frac{5}{2}x - 5=-3x + 17 $.
  - Multiply by 2: $ 5x-10=-6x + 34 $.
  - Add $ 6x $ to both sides: $ 11x-10 = 34 $.
  - Add 10 to both sides: $ 11x = 44 $.
  - Divide by 11: $ x = 4 $.
  - Substitute $ x = 4 $ into $ y=-3x + 17 $, $ y=-12 + 17 = 5 $.

# Answer:
$(4, 5)$

### Part 2: Solve $ f(x)=300 - 10x $ and $ g(x)=75 + 15x $ graphically for $ f(x)=g(x) $
# Explanation:
## Step 1: Rewrite as equations
We need to solve $ y = 300 - 10x $ and $ y=75 + 15x $.

## Step 2: Find intercepts for $ y = 300 - 10x $
- For $ x = 0 $, $ y = 300 $, y - intercept $ (0, 300) $.
- For $ y = 0 $, $ 0=300 - 10x\Rightarrow10x = 300\Rightarrow x = 30 $, x - intercept $ (30, 0) $.

## Step 3: Find intercepts for $ y=75 + 15x $
- For $ x = 0 $, $ y = 75 $, y - intercept $ (0, 75) $.
- For $ y = 0 $, $ 0=75 + 15x\Rightarrow15x=-75\Rightarrow x=-5 $, x - intercept $ (-5, 0) $.

## Step 4: Find intersection
- Solve algebraically: $ 300 - 10x=75 + 15x $.
  - Add $ 10x $ to both sides: $ 300=75 + 25x $.
  - Subtract 75 from both sides: $ 225 = 25x $.
  - Divide by 25: $ x = 9 $.

# Answer:
$ x = 9 $

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QUESTION IMAGE

Question

Explanation:

Part 1a: Solve \( y = x - 6 \) and \( y = -\frac{1}{2}x - 3 \) graphically

Step 1: Find intercepts for \( y = x - 6 \)

Step 2: Find intercepts for \( y = -\frac{1}{2}x - 3 \)

Step 3: Graph the lines

Step 1: Find intercepts for \( y=-2x - 8 \)

Step 2: Find intercepts for \( y=\frac{2}{3}x + 8 \)

Step 3: Graph the lines and find intersection

Step 1: Find intercepts for \( y=\frac{5}{4}x - 4 \)

Step 2: Find intercepts for \( y = -\frac{1}{2}x+3 \)

Step 3: Find intersection

Answer:

Part 1b: Solve \( y = -2x - 8 \) and \( y=\frac{2}{3}x + 8 \) graphically