B. Determine if the tables and graphs below express a direct variation between the va
If they do, find the constant of variation and an equation that defines the relation.
1.
2
2.
6.
3.
5.
X
y
X
X
y
X
X
y
X
y
y
y
1
-3
7
3
2
6
7
-21
6 -9
-15 10 -20 25
-3 2
5
3
2
16
20
12 15
3
-9
8
9
4
– 12
4
3
-28
-12
5
4
24 28
32
18 21 24
10
12
11 13
Answer:
To determine if the tables and graphs express a direct variation, we need to check if the ratios of y to x are constant. Let’s calculate these ratios for each set of values:
1. \( \frac{-3}{1} = -3 \)
2. \( \frac{6}{2} = 3 \)
3. \( \frac{-21}{7} = -3 \)
4. \( \frac{-9}{3} = -3 \)
5. \( \frac{10}{2} = 5 \)
6. \( \frac{-15}{-5} = 3 \)
Since the ratios are constant (-3, 3, -3, -3, 5, 3), we can conclude that there is a direct variation.
Now, let’s find the constant of variation (k). It’s the common ratio in the direct variation equation \(y = kx\).
For example, using the first set: \( -3 = k \times 1 \), so \( k = -3 \).
Thus, the equation defining the relation is \(y = -3x\).
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