Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Rational Equations

College Algebra

1. Equations & Inequalities

Rational Equations

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1:20 minutes

Problem 3

Textbook Question

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

Identify the direct variation equation given: \( y = 6x \).

Understand that in direct variation, \( y \) is directly proportional to \( x \), meaning \( y = kx \) where \( k \) is the constant of proportionality.

Recognize that the constant \( k \) is already given as 6 in the equation \( y = 6x \).

Substitute \( x = 10 \) into the equation to find the new value of \( y \): \( y = 6 \times 10 \).

Calculate the expression \( 6 \times 10 \) to find the value of \( y \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Direct Variation

Direct variation describes a relationship between two variables where one variable is a constant multiple of the other. In the equation y = kx, k is the constant of variation. This means that as x increases, y increases proportionally, and vice versa. Understanding this concept is crucial for solving problems involving proportional relationships.

Substituting Values

Substituting values involves replacing a variable in an equation with a specific number to find the value of another variable. In the context of the equation y = 6x, substituting x with a known value allows us to calculate the corresponding value of y. This technique is essential for solving equations and understanding how changes in one variable affect another.

Proportional Relationships

Proportional relationships occur when two quantities maintain a constant ratio. In the equation y = 6x, the ratio of y to x is always 6, indicating that for every unit increase in x, y increases by 6 units. Recognizing these relationships helps in predicting values and understanding the behavior of linear equations.