8.9 Use Direct and Inverse Variation

Direct and Inverse Variation

Direct and inverse variation describe how two quantities are connected. In direct variation, both quantities move in the same direction. In inverse variation, they move in opposite directions. These patterns show up constantly in science, finance, and everyday problem-solving, and they tie directly into the rational expressions you've been working with in this unit.

Direct and Inverse Variation

Direct variation problem solving

Direct variation means two quantities $x$ and $y$ are related by the equation:

$y = kx$

Here, $k$ is a non-zero constant called the constant of variation (or constant of proportionality). The key idea: when $x$ doubles, $y$ doubles. When $x$ triples, $y$ triples. They scale together.

You can spot direct variation by checking whether the ratio $\frac{y}{x}$ stays the same for every pair of values. If it does, you have direct variation, and that constant ratio is $k$.

How to solve a direct variation problem:

1. Write the equation $y = kx$
2. Substitute a known pair of $x$ and $y$ values to solve for $k$
3. Rewrite the equation with your value of $k$
4. Substitute the new $x$ (or $y$) to find the unknown quantity

Example: The cost of apples ($y$) varies directly with the number of pounds purchased ($x$). If 3 pounds cost $6, find the cost of 5 pounds.

1. Start with $y = kx$
2. Plug in $y = 6$ and $x = 3$: $6 = k(3)$, so $k = 2$
3. The equation is $y = 2x$
4. For 5 pounds: $y = 2(5) = 10$. The cost is $10.

Since $y = kx$ is a linear equation with no added constant, the graph of a direct variation is always a straight line passing through the origin $(0, 0)$. The slope of that line is $k$.

Applications of inverse variation

Inverse variation means two quantities $x$ and $y$ are related by the equation:

$y = \frac{k}{x}$

This time, as one quantity increases, the other decreases. You can spot inverse variation by checking whether the product $xy$ stays the same for every pair of values. If it does, that constant product is $k$.

How to solve an inverse variation problem:

1. Write the equation $y = \frac{k}{x}$
2. Substitute a known pair of $x$ and $y$ values to solve for $k$
3. Rewrite the equation with your value of $k$
4. Substitute the new $x$ (or $y$) to find the unknown quantity

Example 1: The time to fill a pool ($y$) is inversely proportional to the number of hoses used ($x$). If it takes 6 hours with 2 hoses, how long with 3 hoses?

1. Start with $y = \frac{k}{x}$
2. Plug in $y = 6$ and $x = 2$: $6 = \frac{k}{2}$, so $k = 12$
3. The equation is $y = \frac{12}{x}$
4. For 3 hoses: $y = \frac{12}{3} = 4$. It takes 4 hours.

Example 2: Light intensity ($y$) is inversely proportional to the square of the distance ($x$) from the source: $y = \frac{k}{x^2}$. If the intensity is 100 lux at 2 meters, find the intensity at 4 meters.

1. Start with $y = \frac{k}{x^2}$
2. Plug in: $100 = \frac{k}{2^2} = \frac{k}{4}$, so $k = 400$
3. The equation is $y = \frac{400}{x^2}$
4. At 4 meters: $y = \frac{400}{16} = 25$ lux

Note that Example 2 is technically an inverse square variation, not a simple inverse variation. Your course may or may not cover this distinction, but the solving process is the same.

Direct vs inverse relationships

Being able to tell these apart is half the battle. Here's a quick comparison:

Recognizing direct variation in context:

• Distance traveled varies directly with time at constant speed. If a car travels 120 miles in 2 hours, then $k = 60$ mph, and in 3 hours it travels $60 \times 3 = 180$ miles.
• Pay varies directly with hours worked at a constant rate. If someone earns $75 for 5 hours, then $k = 15$ dollars/hour, and 8 hours earns $15 \times 8 = 120$ dollars.

Recognizing inverse variation in context:

• Days to complete a project varies inversely with the number of workers. If 6 workers finish in 10 days, then $k = 60$, and 4 workers would take $\frac{60}{4} = 15$ days.
• Boyle's Law: gas pressure varies inversely with volume at constant temperature. If pressure is 2 atm at 5 liters, then $k = 10$, and at 2 liters the pressure is $\frac{10}{2} = 5$ atm.

A practical tip: ask yourself, "If I increase one quantity, does the other go up or down?" Same direction means direct. Opposite directions means inverse.

Graphical representation and analysis

The graph of direct variation ($y = kx$) is a straight line through the origin. If $k > 0$, the line rises from left to right. If $k < 0$, it falls. The steeper the line, the larger the absolute value of $k$.

The graph of inverse variation ($y = \frac{k}{x}$) is a hyperbola. The curve approaches both axes but never touches them (the axes are asymptotes). For positive $k$, the curve sits in the first and third quadrants. As $x$ gets very large, $y$ gets closer and closer to zero, but never reaches it.

When you're given a graph on a test, check two things:

• Does it pass through the origin? If yes, it could be direct variation.
• Does it curve toward the axes without crossing them? That's the signature shape of inverse variation.

You can also use graphs to estimate unknown values by reading coordinates off the curve or line, then verify with the equation.