91Ó°ÊÓ

The variation constant, often symbolized as \( k \), plays a crucial role in direct variation equations. It connects the two variables in such a way that one is always a multiple of the other. In our problem, when we know that \( y = 3 \) when \( x = 33 \), we can find the variation constant by using these values to solve the equation. First, we substitute the known values into the direct variation formula \( y = kx \), resulting in \[ 3 = k \times 33 \]. Next, we solve for \( k \) by isolating it on one side. This gives us \[ k = \frac{3}{33} = \frac{1}{11} \]. This constant essentially tells us how y changes relative to x.

The direct variation formula establishes a direct relationship between two variables. It's expressed as \( y = kx \), where \( y \) and \( x \) are the variables, and \( k \) is the variation constant. This formula implies that for every unit increase in \( x\), \( y \) increases by a multiple of \( k\). In our specific example, we used the values provided (\( y = 3 \) and \( x = 33 \)). When we input these values, we get \[ 3 = k \times 33 \], allowing us to solve for \( k\). This leads us directly to the equation \[ y = \frac{1}{11}x \], showing the straightforward linear relationship between \( x\) and \( y \).

Solving equations in the context of direct variation typically involves a few straightforward steps. Initially, you substitute the known values into the direct variation formula \( y = kx \). Given our example, we substituted \( y = 3 \) and \( x = 33 \) into the formula, creating the equation \[ 3 = k \times 33 \]. To find the variation constant \( k \), we rearranged the equation to isolate \( k \) on one side. This meant dividing both sides by 33, resulting in \( k = \frac{3}{33} = \frac{1}{11} \). After finding \( k \), we substituted it back into the direct variation formula, giving us the final equation of variation: \[ y = \frac{1}{11}x \]. Breaking down equations in this manner can make complex problems much more manageable.