To determine if y varies directly with x, we first need to understand the concept of direct variation. A variable y is said to vary directly with x if there is a constant k such that:
y = kx
In this case, the equation we are considering is:
5y = 5x + 10
To check for direct variation, we can rearrange this equation into a more standard form. First, let’s solve for y:
5y = 5x + 10
Divide through by 5:
y = x + 2
Now, we have expressed y in terms of x. The equation y = x + 2 shows that for every increase in x, y increases by the same amount plus an additional 2. This indicates that y does not vary directly with x since a true direct variation would not contain a constant term like +2. In direct variation, when x is zero, y would also be zero.
To find the constant of variation, k, we need to express y in the form of y = kx. In our current equation, we see an additional term (+2), which confirms the absence of a direct variation. Therefore, we cannot find a constant of variation k as required in a direct variation scenario.
In conclusion, since the equation includes a constant term, we can conclude:
- y does not vary directly with x.
- There is no constant of variation k.