Asked by Lee Hooni 🖤
x y
-2 80
-1 70
2 40
5 10
Answer the questions using the table.
(3 points)
the rate of change =
the y−intercept =
the equation of the line in slope intercept form is
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-2 80
-1 70
2 40
5 10
Answer the questions using the table.
(3 points)
the rate of change =
the y−intercept =
the equation of the line in slope intercept form is
Skip to navigation
Answers
Answered by
BTS ARMY
Domain considering the scenario: Justin has $10 in his account and plans to save $25 each week until he reaches $135 for the new game he wants to buy.
Answered by
GPT-4o mini
To analyze the data in the table, we can determine the rate of change (slope), the y-intercept, and form an equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
1. **Rate of change (slope):**
The rate of change (slope) can be calculated using two points from the table. Let's use the points (-1, 70) and (2, 40):
\[
\text{slope} = m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points (x₁, y₁) = (-1, 70) and (x₂, y₂) = (2, 40):
\[
m = \frac{40 - 70}{2 - (-1)} = \frac{-30}{2 + 1} = \frac{-30}{3} = -10
\]
So, the rate of change (slope) is **-10**.
2. **Y-intercept:**
To find the y-intercept, we can use the point-slope form of the equation. We know the slope is -10 and can use one of the points, say (2,40):
Using the slope-intercept formula \(y = mx + b\):
\[
40 = -10(2) + b
\]
\[
40 = -20 + b
\]
\[
b = 40 + 20 = 60
\]
Therefore, the y-intercept is **60**.
3. **Equation of the line in slope-intercept form:**
Now that we have the slope (m = -10) and the y-intercept (b = 60), we can write the equation:
\[
y = -10x + 60
\]
In summary:
- The rate of change = **-10**
- The y-intercept = **60**
- The equation of the line in slope-intercept form is **y = -10x + 60**.
1. **Rate of change (slope):**
The rate of change (slope) can be calculated using two points from the table. Let's use the points (-1, 70) and (2, 40):
\[
\text{slope} = m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points (x₁, y₁) = (-1, 70) and (x₂, y₂) = (2, 40):
\[
m = \frac{40 - 70}{2 - (-1)} = \frac{-30}{2 + 1} = \frac{-30}{3} = -10
\]
So, the rate of change (slope) is **-10**.
2. **Y-intercept:**
To find the y-intercept, we can use the point-slope form of the equation. We know the slope is -10 and can use one of the points, say (2,40):
Using the slope-intercept formula \(y = mx + b\):
\[
40 = -10(2) + b
\]
\[
40 = -20 + b
\]
\[
b = 40 + 20 = 60
\]
Therefore, the y-intercept is **60**.
3. **Equation of the line in slope-intercept form:**
Now that we have the slope (m = -10) and the y-intercept (b = 60), we can write the equation:
\[
y = -10x + 60
\]
In summary:
- The rate of change = **-10**
- The y-intercept = **60**
- The equation of the line in slope-intercept form is **y = -10x + 60**.
Answered by
GPT-4o mini
To determine the domain of Justin's savings scenario, we first need to understand the context and establish the variables involved.
1. **Initial Conditions:**
- Justin starts with **$10** in his account.
- He plans to save **$25** each week.
- He wants to reach a total of **$135**.
2. **Setting Up the Equation:**
Let \( x \) represent the number of weeks he saves.
The total amount \( A \) in his account after \( x \) weeks can be expressed as:
\[
A = 10 + 25x
\]
Justin wants this total amount \( A \) to be at least **$135** to buy the game:
\[
10 + 25x \geq 135
\]
3. **Solving the Inequality:**
Subtracting 10 from both sides gives:
\[
25x \geq 125
\]
Dividing by 25:
\[
x \geq 5
\]
This means that Justin needs to save for at least **5 weeks** to reach his goal.
4. **Determining the Domain:**
Since \( x \) represents weeks, and he can't save for a negative number of weeks, the natural boundaries for \( x \) would be:
- **Lower Bound:** \( x \geq 0\) (he can start from week 0)
- **Upper Bound:** There is no upper limit since he can continue to save even after reaching $135.
Considering the context:
- The lowest he can start is week 0.
- He needs at least 5 weeks to save enough money.
Thus, the valid domain for \( x \) in this scenario is:
\[
[5, \infty)
\]
In conclusion, the domain of the number of weeks \( x \) that Justin can save until he reaches his goal is **\[5, \infty\)**.
1. **Initial Conditions:**
- Justin starts with **$10** in his account.
- He plans to save **$25** each week.
- He wants to reach a total of **$135**.
2. **Setting Up the Equation:**
Let \( x \) represent the number of weeks he saves.
The total amount \( A \) in his account after \( x \) weeks can be expressed as:
\[
A = 10 + 25x
\]
Justin wants this total amount \( A \) to be at least **$135** to buy the game:
\[
10 + 25x \geq 135
\]
3. **Solving the Inequality:**
Subtracting 10 from both sides gives:
\[
25x \geq 125
\]
Dividing by 25:
\[
x \geq 5
\]
This means that Justin needs to save for at least **5 weeks** to reach his goal.
4. **Determining the Domain:**
Since \( x \) represents weeks, and he can't save for a negative number of weeks, the natural boundaries for \( x \) would be:
- **Lower Bound:** \( x \geq 0\) (he can start from week 0)
- **Upper Bound:** There is no upper limit since he can continue to save even after reaching $135.
Considering the context:
- The lowest he can start is week 0.
- He needs at least 5 weeks to save enough money.
Thus, the valid domain for \( x \) in this scenario is:
\[
[5, \infty)
\]
In conclusion, the domain of the number of weeks \( x \) that Justin can save until he reaches his goal is **\[5, \infty\)**.