Question

this table shows the relationship between the length, l, and the width, w, of a rectangle
l | 5 | 8 | 11 | 14 | 17
w | 1 | 4 | 7 | 10 | 13
select from the drop - down menu to correctly complete the statement
the value of the dependent variable w is always choose the value of the independent variable l

Explanation:

Step1: Analyze the relationship

We have pairs of values for \( l \) (length) and \( w \) (width). Let's check the difference between \( l \) and \( w \) for each pair:
- For \( l = 5 \), \( w = 1 \): \( 5 - 1 = 4 \)
- For \( l = 8 \), \( w = 4 \): \( 8 - 4 = 4 \)
- For \( l = 11 \), \( w = 7 \): \( 11 - 7 = 4 \)
- For \( l = 14 \), \( w = 10 \): \( 14 - 10 = 4 \)
- For \( l = 17 \), \( w = 13 \): \( 17 - 13 = 4 \)

So the relationship is \( w = l - 4 \), which means \( w \) depends on \( l \). So \( w \) is the dependent variable and \( l \) is the independent variable. Now we check the difference between \( w \) and \( l \), but actually, we can see the pattern of how \( w \) relates to \( l \). Wait, the question is about the value of the dependent variable \( w \) compared to the independent variable \( l \). From the differences above, \( w = l - 4 \), so \( w \) is always less than \( l \) (since we subtract 4 from \( l \) to get \( w \)). Wait, but let's check the numbers. Wait, the drop - down menu options (from the image, the numbers on the left are 1 - 10, but let's re - evaluate. Wait, the pairs are (5,1), (8,4), (11,7), (14,10), (17,13). Let's see the difference between \( l \) and \( w \): \( 5 - 1=4 \), \( 8 - 4 = 4 \), \( 11 - 7=4 \), \( 14 - 10 = 4 \), \( 17 - 13=4 \). So \( w=l - 4 \), which means \( w \) is always 4 less than \( l \)? Wait, no, the question is "The value of the dependent variable \( w \) is always [ ] the value of the independent variable \( l \)". Let's check the values:
- When \( l = 5 \), \( w = 1 \), \( 1<5 \)
- When \( l = 8 \), \( w = 4 \), \( 4<8 \)
- When \( l = 11 \), \( w = 7 \), \( 7<11 \)
- When \( l = 14 \), \( w = 10 \), \( 10<14 \)
- When \( l = 17 \), \( w = 13 \), \( 13<17 \)

Wait, but also, let's check the difference. Wait, maybe the relationship is linear. The slope between \( l \) and \( w \): Let's take two points \((l_1,w_1)=(5,1)\) and \((l_2,w_2)=(8,4)\). The slope \( m=\frac{w_2 - w_1}{l_2 - l_1}=\frac{4 - 1}{8 - 5}=\frac{3}{3}=1 \). And the y - intercept (if we consider \( w \) as \( y \) and \( l \) as \( x \)) is \( w= l + b \), plugging in \( l = 5 \), \( w = 1 \): \( 1=5 + b\), so \( b=-4 \). So \( w=l - 4 \). So \( w \) is always 4 less than \( l \), or we can say \( w \) is always less than \( l \), but looking at the numbers, the difference is 4. Wait, the drop - down menu on the left has numbers 1 - 10, and from the pairs, when \( l \) increases by 3, \( w \) increases by 3. Wait, maybe the question is about the value of \( w \) compared to \( l \) in terms of the arithmetic operation. Wait, the key is to find the relationship. Since \( w=l - 4 \), then \( w \) is always 4 less than \( l \), but looking at the numbers, for example, when \( l = 5 \), \( w = 1 \) (5 - 4 = 1), \( l = 8 \), \( w = 4 \) (8 - 4 = 4), etc. So the value of \( w \) is always 4 less than \( l \), but if we look at the numbers, the difference between \( l \) and \( w \) is 4, but the question is "The value of the dependent variable \( w \) is always [ ] the value of the independent variable \( l \)". From the numbers, \( w \) is always 4 less than \( l \), but if we check the options (the vertical list on the left: 1,2,3,4,5,6,7,8,9,10), and the relationship \( w=l - 4 \), so for each \( l \), \( w = l-4 \), which means \( w \) is always 4 less than \( l \), but also, if we look at the values, \( w \) is always less than \( l \), but the specific number? Wait, no, the question is probably asking for the operation or the relationship. Wait, maybe I misread. Let's re - examine the table:

| \( l \) | 5 | 8 | 11 | 14 | 17 |
| --- | --- | --- | --- | --- | --- |
| \( w \) | 1 | 4 | 7 | 10 | 13 |

Now, let's find the difference between \( l \) and \( w \): \( 5 - 1 = 4 \), \( 8 - 4 = 4 \), \( 11 - 7 = 4 \), \( 14 - 10 = 4 \), \( 17 - 13 = 4 \). So \( w=l - 4 \). So the value of the dependent variable \( w \) is always 4 less than the value of the independent variable \( l \). But looking at the drop - down menu on the left (the numbers 1 - 10), and the relationship, we can see that \( w = l-4 \), so for example, when \( l = 5 \), \( w = 1 \) (5 - 4 = 1), \( l = 8 \), \( w = 4 \) (8 - 4 = 4), etc. So the value of \( w \) is always 4 less than \( l \), but if we consider the numbers, the difference is 4, so the value of \( w \) is always 4 less than \( l \), which means \( w = l - 4 \), so the answer is that \( w \) is always 4 less than \( l \), but from the drop - down menu (the vertical list with numbers 1 - 10), and the pairs, we can see that the difference is 4, so the value of \( w \) is always 4 less than \( l \), so the correct value to choose from the drop - down (the number 4? Wait, no, the drop - down on the left has numbers 1 - 10, and the key is that \( w=l - 4 \), so \( w \) is always 4 less than \( l \), so the answer is that the value of \( w \) is always 4 less than \( l \), but in terms of the numbers, when we look at the relationship, the difference is 4. Wait, maybe the question is "The value of the dependent variable \( w \) is always [ ] the value of the independent variable \( l \)", and the options are numbers, and from the pairs, when \( l \) is 5, \( w \) is 1 (5 - 4 = 1), \( l = 8 \), \( w = 4 \) (8 - 4 = 4), \( l = 11 \), \( w = 7 \) (11 - 4 = 7), etc. So the pattern is \( w=l - 4 \), so \( w \) is always 4 less than \( l \), so the answer is 4 less, but looking at the drop - down menu (the vertical list with numbers 1 - 10), the correct number is 4? Wait, no, the drop - down on the left is for selecting the number, and from the relationship \( w=l - 4 \), the value of \( w \) is always 4 less than \( l \), so the answer is that the value of \( w \) is always 4 less than \( l \), so when choosing from the drop - down, we select 4? Wait, maybe I made a mistake. Let's re - calculate the differences:

For \( l = 5 \), \( w = 1 \): \( 5-1 = 4 \)

For \( l = 8 \), \( w = 4 \): \( 8 - 4=4 \)

For \( l = 11 \), \( w = 7 \): \( 11 - 7 = 4 \)

For \( l = 14 \), \( w = 10 \): \( 14 - 10=4 \)

For \( l = 17 \), \( w = 13 \): \( 17 - 13 = 4 \)

So the difference between \( l \) and \( w \) is always 4, which means \( w=l - 4 \). So the value of the dependent variable \( w \) is always 4 less than the value of the independent variable \( l \). So the answer is that the value of \( w \) is always 4 less than \( l \), and from the drop - down menu (the vertical list with numbers 1 - 10), we select 4? Wait, no, the drop - down is for the value that \( w \) is always [ ] the value of \( l \), and the numbers are 1 - 10. Since \( w=l - 4 \), then \( w \) is always 4 less than \( l \), so the answer is 4.

Step2: Confirm the relationship

We can check all the pairs again. For each pair \((l,w)\), \( l - w=4 \). So \( w = l - 4 \), which implies that the value of \( w \) is always 4 less than the value of \( l \).

Answer:

4