Question

Use the table to answer the question.
x y
-2 -1
0 1
2 3
Write an equation that represents this nonproportional linear relationship.

Answer 1

A family is going from Dallas, Texas, to Los Angeles, California. They have traveled 500 miles so far. The next day, their average speed is 70 miles per hour. Write an equation for the distance they traveled, y, and the hours they have driven, x. Indicate whether this is a proportional or a nonproportional situation.

Answer 2

You are baking a cake. For every three cups of flour, you need to add two eggs. Write an equation for the relationship between the number of eggs, y, and the amount of flour, x. Distinguish whether this is a proportional or nonproportional situation.

Answer 3

To find the equation of the nonproportional linear relationship represented by the given points, we first need to determine the slope of the line defined by the points \((-2, -1)\), \((0, 1)\), and \((2, 3)\).

1. **Calculate the slope (m)** using any two points, for example, \((0, 1)\) and \((2, 3)\):

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 1}{2 - 0} = \frac{2}{2} = 1
\]

2. **Use the point-slope form of the equation**. We can use the point \((0, 1)\) (where \(y\) intercept \(b = 1\)) to formulate the equation:

The slope-intercept form is:

\[
y = mx + b
\]

Substituting \(m = 1\) and \(b = 1\):

\[
y = 1x + 1
\]

or simply:

\[
y = x + 1
\]

Thus, the equation that represents the nonproportional linear relationship is:

\[
y = x + 1
\]

Answer 4

To write an equation for the distance they have traveled, \(y\), based on the number of hours they have driven, \(x\), we need to consider both the distance they have already traveled (500 miles) and the additional distance they will travel at an average speed of 70 miles per hour.

1. **Initial travel distance:** 500 miles (this can be considered as a fixed amount added to the travel)
2. **Distance traveled after additional hours:** 70 miles per hour multiplied by \(x\) hours.

Thus, the equation for the total distance \(y\) can be formulated as:

\[
y = 500 + 70x
\]

### Identifying Proportional or Nonproportional:

- In a proportional relationship, the equation is of the form \(y = kx\) (where \(k\) is a constant), and it passes through the origin (0,0).
- In this equation, because of the \(+500\), the line does not pass through the origin. The distance \(y\) at \(x = 0\) is 500 miles.

This means that:

- The relationship is **nonproportional** because there is a constant amount added (the initial distance), making it not solely dependent on \(x\).

### Conclusion:

The equation representing the distance they have traveled is:

\[
y = 500 + 70x
\]

This represents a **nonproportional** situation.

Answer 5

To express the relationship between the number of eggs \(y\) and the amount of flour \(x\), we can start by identifying the ratio given in the problem:

- For every **3 cups of flour**, you need to add **2 eggs**.

From this ratio, we can see that the relationship can be set up using the concept of proportions. We can define the equation based on the ratio of eggs to flour.

First, let's express this ratio mathematically:

\[
\frac{y}{x} = \frac{2}{3}
\]

To find the relationship between \(y\) and \(x\), we can rearrange this to get an equation in the form of \(y = kx\), where \(k\) is a constant.

To do that, multiply both sides by \(x\):

\[
y = \frac{2}{3}x
\]

### Identifying Proportional or Nonproportional:

- In this equation \(y = \frac{2}{3}x\), the relationship is linear, and it goes through the origin (0,0) because when \(x = 0\), \(y = 0\) as well.
- This indicates that the number of eggs is directly proportional to the amount of flour, meaning there is a constant ratio between the two quantities.

### Conclusion:

The equation that represents the relationship between the number of eggs \(y\) and the amount of flour \(x\) is:

\[
y = \frac{2}{3}x
\]

This is a **proportional** relationship since it maintains a constant ratio and passes through the origin.