sketch the graph of each line. use the x and y intercepts to graph...
13) 5x + 2y = 10
14) 2x + y = -1
identify the rate of change... (the rate of change is constant for this table)
15)
| number of balloons | total cost of balloons (in dollars) |
| --- | --- |
| 2 | 6 |
| 4 | 12 |
| 6 | 18 |
| 8 | 24 |
16)
| hours spent babysitting | money earned ($) |
| --- | --- |
| 1 | 10 |
| 3 | 30 |
| 5 | 50 |
| 7 | 70 |
17) concert tickets cost $15 for adults and $9 for students. the total ticket sale was $4,500. write an equations in standard form that models this situation.
18) mimi wants to spend a $25 gift card on two kinds of in - app purchases for her favorite game: shields that cost $2.99, and tools that cost $4.99. write an equation that models the shields and tools that mimi can afford with the gift card.

Question

Accepted Answer

### Problem 13: $ 5x + 2y = 10 $ (Graphing using intercepts)
# Explanation:
## Step 1: Find x-intercept (set $ y = 0 $)
Substitute $ y = 0 $ into $ 5x + 2y = 10 $:
$ 5x + 2(0) = 10 $
$ 5x = 10 $
$ x = \frac{10}{5} = 2 $
So, x-intercept is $ (2, 0) $.

## Step 2: Find y-intercept (set $ x = 0 $)
Substitute $ x = 0 $ into $ 5x + 2y = 10 $:
$ 5(0) + 2y = 10 $
$ 2y = 10 $
$ y = \frac{10}{2} = 5 $
So, y-intercept is $ (0, 5) $.

## Step 3: Graph the line
Plot the points $ (2, 0) $ (on the x-axis) and $ (0, 5) $ (on the y-axis), then draw a straight line through them.

# Answer:
x-intercept: $ (2, 0) $, y-intercept: $ (0, 5) $; line connects these two points.

### Problem 14: $ 2x + y = -1 $ (Graphing using intercepts)
# Explanation:
## Step 1: Find x-intercept (set $ y = 0 $)
Substitute $ y = 0 $ into $ 2x + y = -1 $:
$ 2x + 0 = -1 $
$ 2x = -1 $
$ x = -\frac{1}{2} = -0.5 $
So, x-intercept is $ (-0.5, 0) $.

## Step 2: Find y-intercept (set $ x = 0 $)
Substitute $ x = 0 $ into $ 2x + y = -1 $:
$ 0 + y = -1 $
$ y = -1 $
So, y-intercept is $ (0, -1) $.

## Step 3: Graph the line
Plot the points $ (-0.5, 0) $ and $ (0, -1) $, then draw a straight line through them.

# Answer:
x-intercept: $ (-0.5, 0) $, y-intercept: $ (0, -1) $; line connects these two points.

### Problem 15: Rate of change (Balloon table)
# Explanation:
Rate of change (slope) for a table is $ \frac{\text{Change in } y}{\text{Change in } x} $.
Let $ x $ = Number of Balloons, $ y $ = Total Cost (dollars).

Take two points: $ (2, 6) $ and $ (4, 12) $.
Change in $ y $: $ 12 - 6 = 6 $
Change in $ x $: $ 4 - 2 = 2 $
Rate of change: $ \frac{6}{2} = 3 $.

Verify with another pair: $ (4, 12) $ and $ (6, 18) $.
Change in $ y $: $ 18 - 12 = 6 $
Change in $ x $: $ 6 - 4 = 2 $
Rate of change: $ \frac{6}{2} = 3 $. Consistent.

# Answer:
Rate of change is $ 3 $ dollars per balloon.

### Problem 16: Rate of change (Babysitting table)
# Explanation:
Rate of change (slope) for a table is $ \frac{\text{Change in } y}{\text{Change in } x} $.
Let $ x $ = Hours Spent, $ y $ = Money Earned (\$).

Take two points: $ (1, 10) $ and $ (3, 30) $.
Change in $ y $: $ 30 - 10 = 20 $
Change in $ x $: $ 3 - 1 = 2 $
Rate of change: $ \frac{20}{2} = 10 $.

Verify with another pair: $ (3, 30) $ and $ (5, 50) $.
Change in $ y $: $ 50 - 30 = 20 $
Change in $ x $: $ 5 - 3 = 2 $
Rate of change: $ \frac{20}{2} = 10 $. Consistent.

# Answer:
Rate of change is $ 10 $ dollars per hour.

### Problem 17: Concert ticket equation
# Explanation:
Let $ a $ = number of adult tickets, $ s $ = number of student tickets.

Adult cost: $ \$15 $ per ticket, so total adult revenue: $ 15a $.
Student cost: $ \$9 $ per ticket, so total student revenue: $ 9s $.
Total ticket sales: $ \$4500 $.

Equation: $ 15a + 9s = 4500 $ (standard form: $ Ax + By = C $).

# Answer:
$ 15a + 9s = 4500 $

### Problem 18: Gift card equation
# Explanation:
Let $ s $ = number of shields, $ t $ = number of tools.

Cost of shields: $ \$2.99 $ per shield, so total shield cost: $ 2.99s $.
Cost of tools: $ \$4.99 $ per tool, so total tool cost: $ 4.99t $.
Total cost must be $ \leq \$25 $ (since she uses a \$25 gift card).

Equation: $ 2.99s + 4.99t = 25 $ (or $ \leq 25 $, but standard form for "afford" is equality for maximum).

# Answer:
$ 2.99s + 4.99t = 25 $ (or $ 2.99s + 4.99t \leq 25 $)

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QUESTION IMAGE

Question

Explanation:

Problem 13: \( 5x + 2y = 10 \) (Graphing using intercepts)

Step 1: Find x-intercept (set \( y = 0 \))

Step 2: Find y-intercept (set \( x = 0 \))

Step 3: Graph the line

Step 1: Find x-intercept (set \( y = 0 \))

Step 2: Find y-intercept (set \( x = 0 \))

Step 3: Graph the line

Answer:

Problem 14: \( 2x + y = -1 \) (Graphing using intercepts)