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 to find the constant \(k\). Substitute \(x = 5\) and \(y = 30\) into the equation \(y = kx\) to get \$30 = k \times 5$.
3. Solve for \(k\) by dividing both sides of the equation by 5, resulting in \(k = \frac{30}{5}\).
4. Write the complete direct variation equation using the found value of \(k\), which will be \(y = kx\) with the specific \(k\) value.
5. Find the value of \(y\) when \(x = 10\) by substituting \(x = 10\) into the equation \(y = kx\) and calculating \(y\).

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable is a constant multiple of another. In the equation y = kx, y varies directly as x, meaning if x changes, y changes proportionally by the constant k.

Constant of Variation

The constant of variation (k) is the fixed multiplier in a direct variation equation y = kx. It can be found by substituting known values of x and y, allowing you to determine the exact relationship between the variables.

Substitution to Find Unknown Values

Once the constant of variation is known, substitution involves replacing the variable x with a given value to calculate the corresponding y. This method helps solve for unknowns in direct variation problems.