How to Solve Inverse Variation?
Inverse variation indicates an inverse relationship between two quantities. In this step-by-step guide, you will learn more about inverse variation.
Solve Inverse Variation: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Identify the inputFind the x-value, expression, or inner function being used.
- Apply the ruleSubstitute with parentheses so signs and powers stay clear.
- Interpret the outputState the value, point, interval, domain, range, or inverse relationship.
Worked examples
Evaluate a function
- Replace x with 2.
- Compute 4(2) – 3.
- Simplify.
Compose functions
- Find g(3) = 6.
- Use that as the input for f.
- f(6) = 7.
Try one before moving on
Solve Inverse Variation: pop-up practice
A step-by-step guide to inverse variation
There are two types of proportionalities. These are direct variation and inverse variation. Two non-zero quantities have an inverse variation if their product yields a constant term (constant of proportionality). In other words, if one quantity is directly proportional to the reciprocal of the other quantity, the two quantities follow inverse variations. This means that an increase in one value leads to a decrease in the other value while a decrease in one value leads to an increase in the other value.
Inverse variation formula
The symbol “\(∝\)” is used to indicate proportionality. If two quantities x and y follow an inverse variation, they are represented as:
\(\color{blue}{x\:∝\:\frac{1}{y}\:or\:y\:∝\:\frac{1}{x}}\)
To convert this expression into an equation, a constant or proportionality factor must be introduced. Therefore, the inverse variation formula is presented as follows:
\(\color{blue}{x=\frac{k}{y}\:\:or\:y=\frac{k}{x}}\)
Here, \(k\) is the constant of proportionality. Also, \(x ≠ 0\) and \(y ≠ 0\).
Product rule for inverse variation
Suppose the two solutions of inverse variation are \((x_1, y_1)\) and \((x_2, y_2)\). This can also be expressed as \(x_1y_1=k\) and \(x_2 y_2=k\).
Using these two equations:
\(x_1y_1 = x_2y_2\) and \(\frac{x_1}{x_2}=\frac{y_2}{y_1}\).
This is the product rule for inverse variation.
Inverse variation graph
The graph of an inverse variation is a rectangular hyperbola. If there are two quantities \(x\) and \(y\) are in inverse variation then their product will be equal to a constant \(k\). Since neither \(x\) nor \(y\) can be zero, the graph never crosses the \(x\)-axis or the \(y\)-axis.
Inverse Variation – Example 1:
If \(x = 15\) and \(y = 4\) follow an inverse variation, find the constant of proportionality.
Solution: Since \(x\) and \(y\) are in inverse variation therefore, \(xy=k\):
\(k= 15\times 4=60\)
Related to This Article
More math articles
- 10 Most Common 8th Grade MCAS Math Questions
- 4th Grade IAR Math Practice Test Questions
- The Practical Benefits Of Knowing Math – The 10 Best Industries For Math Geeks
- Texas STAAR Algebra 1 Free Worksheets: 64 Printable TEKS-Aligned Algebra 1 PDFs with Answer Keys
- Maryland MCAP Grade 3 Math Worksheets — 49 Free Printable Single-Skill PDFs
- Evaluating Variables and Expressions
- DAT Quantitative Reasoning Math Practice Test Questions
- How to Prepare for the Pre-Algebra Test?
- How to Master the Intricacies of the Coordinate Plane
- Online Baccarat vs. Land Baccarat: Where the Math Actually Diverges
What people say about "How to Solve Inverse Variation? - Effortless Math"?
No one replied yet.