Question

Which of the following tables represents a linear relationship that is also proportional?


x 2 3 4
y −3 0 3

x 4 2 0
y −2 −1 0

x −2 1 4
y 0 1 2

x 0 1 2
y −4 0 4

Answer 1

A proportional relationship is a specific type of linear relationship that goes through the origin (0,0). This means that for a relationship to be proportional, if \(x = 0\), then \(y\) must also equal 0.

Let's analyze the given tables one by one to determine if any of them represent a linear relationship that is also proportional:

1. **Table 1:**
```
x: 2 3 4
y: -3 0 3
```
Here, when \(x = 0\), \(y\) is not defined. Since \(y\) does not equal 0 when \(x\) is 0, this relationship is not proportional.

2. **Table 2:**
```
x: 4 2 0
y: -2 -1 0
```
Again, when \(x = 0\), \(y = 0\), which is consistent. However, this set of points does not form a proportion because for the other values, the ratio \(y/x\) is not constant. For example, \(y/x\) for different points will yield different values. Hence, this is not a proportional relationship.

3. **Table 3:**
```
x: -2 1 4
y: 0 1 2
```
For \(x = 0\), \(y\) is not defined, and it also does not pass through the origin, making it not proportional.

4. **Table 4:**
```
x: 0 1 2
y: -4 0 4
```
In this case, \(y = 0\) when \(x = 0\). However, we notice that as \(x\) increases, \(y\) does not keep a consistent ratio. For instance, \(y/x\) cannot be consistently calculated, so it doesn’t form a constant ratio of \(k\). Hence, while it passes through the origin, it's not proportional.

Considering these analyses, none of the tables represent a linear relationship that is also proportional since they either do not pass through the origin or do not have a constant ratio.

**Final Answer:** None of the options represent a linear relationship that is also proportional.