Fill in the blank(s) to correctly complete each sentence, or answer the question as appropriate. In the equation y = 6x, y varies directly as x. When x=5, y=30. What is the value of y when x=10?

Verified step by step guidance

1. Identify the type of variation described: since y varies directly as x, the relationship can be written as \(y = kx\), where \(k\) is the constant of proportionality.
2. Use the given values \(x = 5\) and \(y = 30\) to find the constant \(k\) by substituting into the equation: \(30 = k \times 5\).
3. Solve for \(k\) by dividing both sides of the equation by 5: \(k = \frac{30}{5}\).
4. Now that you have the value of \(k\), write the complete direct variation equation: \(y = kx\) with the found \(k\).
5. To find \(y\) when \(x = 10\), substitute \(x = 10\) into the equation and solve for \(y\): \(y = k \times 10\).

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable is a constant multiple of another, expressed as y = kx. Here, y varies directly as x means y changes proportionally with x, and k is the constant of proportionality.

Constant of Proportionality

The constant of proportionality (k) is the fixed number that relates two variables in direct variation. It can be found by dividing y by x (k = y/x) when values of x and y are known, allowing prediction of y for any x.

Substitution to Find Unknown Values

Once the constant k is known, substitution involves replacing x with a given value in the equation y = kx to find the corresponding y. This method helps solve for unknown variables in direct variation problems.