Example Card: Identifying an Inverse Variation
Model and solve inverse variation problems in Grade 9 Algebra using y = k/x. Find the constant of variation k and use it to predict unknown values. (Saxon Algebra 1, Grade 9)
Key Concepts
Isolating the variable $y$ is the key to determining if a relationship is an inverse variation. This example demonstrates one of the key ideas in this lesson, identifying an inverse variation.
Example Problem Tell whether the relationship $3xy = 21$ is an inverse variation. Explain why.
Step by Step 1. Start with the given equation: $3xy = 21$. 2. To check if this is an inverse variation, we need to see if it can be written in the form $y = \frac{k}{x}$, where $k$ is a non zero constant. 3. First, isolate the product $xy$ by dividing both sides by $3$: $xy = \frac{21}{3}$. 4. Simplify the right side: $xy = 7$. 5. Now, solve for $y$ by dividing both sides by $x$: $y = \frac{7}{x}$. 6. This equation matches the form $y = \frac{k}{x}$, with the constant of variation $k=7$. Therefore, the relationship is an inverse variation.
Common Questions
What is inverse variation in algebra?
Inverse variation describes a relationship where y = k/x for a nonzero constant k. As x increases, y decreases proportionally, and the product xy always equals the constant k. The graph is a hyperbola.
How do you find the constant of variation in an inverse variation?
Multiply any x-value by its corresponding y-value: k = xy. If this product is the same for every pair in the table, the relationship is an inverse variation and k is your constant.
How do you distinguish inverse variation from direct variation?
In direct variation y = kx the ratio y/x is constant, and the graph is a line through the origin. In inverse variation y = k/x the product xy is constant, and the graph is a hyperbola. Check which quantity stays constant to identify the type.