Question

x:
-2
-1
2
5

y:
80
70
40
10

Answer the questions using the table.

the rate of change =

Answer 1

the y−intercept =

Answer 2

the equation of the line in slope intercept form is

Answer 3

Michele has $25. She plans to save $10 per week until she has a total of $75.

Which equation matches this scenario?

options:

y - 55 = 10(x - 3)
y - 55 = 10(x - 3)

10x + y = 25
10x + y = 25

y = 10x - 25
y = 10x - 25

y + 4 = 10 (x - 65)

Answer 4

The equation 7x + 5y = 100 is in standard form. This form tells us the x and y intercepts easily.

What are the x and y intercepts?

options:

The line crosses the x-axis at (20, 0) and crosses the y-axis at (0, 100/7).
The line crosses the x-axis at (20, 0) and crosses the y-axis at (0, 100/7).

The line crosses the x-axis at (7, 0) and crosses the y-axis at (0, 5).
The line crosses the x-axis at (7, 0) and crosses the y-axis at (0, 5).

The line crosses the x-axis at (100/7, 0) and crosses the y-axis at (0, 20).
The line crosses the x-axis at (100/7, 0) and crosses the y-axis at (0, 20).

The line crosses the x-axis at (7/100, 0) and crosses the y-axis at (0, 1/20).
The line crosses the x-axis at (7/100, 0) and crosses the y-axis at (0, 1/20).

Answer 5

Ryan is tracking how much money he has each week.

He started with $60.

He spent $5 per week for the first 6 weeks.

He spent $10 per week after week 6 until the money's all spent.



The graph of the scenario would be ______ because the rate of change changes.

Answer 6

To find the rate of change between the points given in the table, we can use the formula for the slope (rate of change) between two points \((x_1, y_1)\) and \((x_2, y_2)\):

\[
\text{Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's calculate the rate of change for the pairs of points in the table:

1. For the points \((-2, 80)\) and \((-1, 70)\):
\[
\text{Rate of Change} = \frac{70 - 80}{-1 - (-2)} = \frac{-10}{1} = -10
\]

2. For the points \((-1, 70)\) and \((2, 40)\):
\[
\text{Rate of Change} = \frac{40 - 70}{2 - (-1)} = \frac{-30}{3} = -10
\]

3. For the points \((2, 40)\) and \((5, 10)\):
\[
\text{Rate of Change} = \frac{10 - 40}{5 - 2} = \frac{-30}{3} = -10
\]

The rate of change between any and all pairs of points from the provided data is \(-10\).

Thus, the answer is:
**Rate of change = -10**.

Answer 7

To find the y-intercept of the linear relationship between the variables \(x\) and \(y\) represented by the given points, we can use the slope-intercept form of a linear equation:

\[
y = mx + b
\]

where:
- \(m\) is the slope (rate of change we calculated earlier),
- \(b\) is the y-intercept.

Since we found that the slope \(m = -10\), we can use one of the points to solve for \(b\). Let's use the point \((2, 40)\):

Plugging in the coordinates into the equation:

\[
40 = -10(2) + b
\]

Solving for \(b\):

\[
40 = -20 + b
\]
\[
b = 40 + 20
\]
\[
b = 60
\]

Thus, the y-intercept is:

**Y-intercept = 60**.

Answer 8

Given the slope \(m = -10\) and the y-intercept \(b = 60\), the equation of the line in slope-intercept form \(y = mx + b\) can be expressed as:

\[
y = -10x + 60
\]

So, the equation of the line is:

**\(y = -10x + 60\)**.

Answer 9

To formulate the equation based on Michele's savings scenario, we start with the information given:

- Michele currently has $25.
- She plans to save $10 per week.
- She aims to save until she has a total of $75.

We can let:
- \(y\) represent the total amount of money Michele has.
- \(x\) represent the number of weeks she saves.

Initially, Michele has $25. After saving \(10x\) dollars over \(x\) weeks, her total amount can be represented as:

\[
y = 25 + 10x
\]

To find the total amount when she reaches $75, we can set up the equation:

\[
y = 25 + 10x
\]
To find when \(y = 75\):

\[
75 = 25 + 10x
\]
Solving for \(x\):

\[
75 - 25 = 10x
\]
\[
50 = 10x
\]
\[
x = 5
\]

Now, substituting \(y = 75\) into our equation form:

Since Michele's total savings will reach 75 at \(x = 5\) weeks, the equation \(y = 25 + 10x\) fits her savings plan.

Now, we can rewrite this:

\[
y = 10x + 25
\]

None of the options exactly match this form, but let's check the relations in the provided options:

1. **Option**: \(y - 55 = 10(x - 3)\)
- This rearranges to \(y = 10x + 25\).
2. **Option**: \(10x + y = 25\)
- This rearranges to \(y = -10x + 25\).
3. **Option**: \(y = 10x - 25\)
- Incorrect form, as the constant is not accurate.
4. **Option**: \(y + 4 = 10(x - 65)\)
- Would not represent the scenario properly either.

The closest match, which correctly represents the equation derived from her savings and fits our expectations, is:

**Option 1**: \(y - 55 = 10(x - 3)\) (since it rearranges to \(y = 10x + 25\)).

Thus, the best choice is:

**\(y - 55 = 10(x - 3)\)** (which simplifies to the same linear behavior of saving $10 per week).

Answer 10

To find the x-intercept and y-intercept of the equation \(7x + 5y = 100\), we can follow these steps:

### Finding the x-intercept:
The x-intercept occurs when \(y = 0\).

Substituting \(y = 0\) into the equation:

\[
7x + 5(0) = 100 \\
7x = 100 \\
x = \frac{100}{7}
\]

So, the x-intercept is \(\left(\frac{100}{7}, 0\right)\).

### Finding the y-intercept:
The y-intercept occurs when \(x = 0\).

Substituting \(x = 0\) into the equation:

\[
7(0) + 5y = 100 \\
5y = 100 \\
y = \frac{100}{5} = 20
\]

So, the y-intercept is \((0, 20)\).

### Summary of intercepts:
- The x-intercept is \(\left(\frac{100}{7}, 0\right)\).
- The y-intercept is \((0, 20)\).

### Conclusion:
The correct option is:

**The line crosses the x-axis at \((100/7, 0)\) and crosses the y-axis at \((0, 20)\).**

Answer 11

In Ryan's scenario, he experiences two different rates of spending over time.

1. **Initial Spending Phase (Weeks 1-6)**:
- He starts with $60 and spends $5 per week for the first 6 weeks.
- The rate of change (slope) for this phase is \(-5\) (indicating a decrease in total money).

2. **Subsequent Spending Phase (After Week 6)**:
- After week 6, he continues to spend, but now at a rate of $10 per week until he runs out of money.
- The rate of change for this phase is \(-10\).

Because the spending rate changes after week 6, the graph depicting Ryan's total money over time would not be a straight line; instead, it would consist of two linear segments with different slopes.

Thus, you can conclude that the graph of the scenario would be **piecewise linear** or **non-linear** because the rate of change changes.

In short, the graph of the scenario would be **non-linear** (or "piecewise linear") because the rate of change changes.