For the data in the table, does y vary directly with x? If it does, write an equation for the direct variation.
X Y
4 6
8 12
10 15
just divide each y by its x. Is the ratio constant?
If so, that means y/x = k,or y=kx, a direct variation.
To determine if y varies directly with x, we need to check if the ratio of y to x is constant for all values in the table.
Let's calculate the ratios for each pair of (x, y) values:
For the first pair (4, 6), the ratio y/x is 6/4 = 1.5.
For the second pair (8, 12), the ratio y/x is 12/8 = 1.5.
For the third pair (10, 15), the ratio y/x is 15/10 = 1.5.
Since the ratio y/x is the same (1.5) for all the pairs, we can conclude that y varies directly with x.
To write an equation for direct variation, we use the equation y = kx, where k is the constant of variation.
From the table, we can see that when x is 4, y is 6. This gives us the equation 6 = 4k.
Solving for k, we divide both sides of the equation by 4:
6/4 = k
1.5 = k
So, the equation for the direct variation is y = 1.5x.