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In a direct variation, the constant of variation is a crucial element. It is the fixed number that relates two variables in a directly proportional way. This constant is commonly represented by the letter 'k'. When we say that y varies directly as x, we write this relationship as y = kx.

For example, in the given exercise, we know that y = 5 when x = 9. By substituting these values into the equation, we get 5 = 9k. We can solve for k by dividing both sides by 9. This gives us k = 5/9.

This constant of variation (k = 5/9) tells us that for every unit increase in x, y increases by 5/9 units. Understanding the constant of variation helps us create a model of how the two variables interact in practical scenarios.

Solving equations is a fundamental skill in algebra. In this exercise, we are asked to find the constant of variation by solving the equation derived from direct variation.

The key steps include:

• Identifying the variables and their given values.
• Setting up the equation based on the direct variation formula, y = kx.
• Substituting the given values into the equation.
• Solving for the unknown variable, which in this case is k.

In our example, we started with the equation 5 = 9k. To isolate k, we divided both sides by 9, resulting in k = 5/9.

This method can be applied to various types of problems involving direct variation, and mastering it will greatly enhance your problem-solving abilities in algebra.

Mathematical relationships describe how two or more variables interact with each other. Direct variation is one such relationship where one variable is a constant multiple of another.

In the given exercise, we see that y directly varies as x. This forms a linear relationship, represented by the equation y = kx. The constant k determines the steepness and direction of the relationship between y and x.

The equation y = (5/9)x tells us that y increases by 5/9 for each unit increase in x. Understanding these relationships helps us predict how changes in one variable affect another, which is crucial in many fields such as physics, economics, and engineering. By analyzing these relationships, we can make informed decisions and solve real-world problems effectively.