Solving Inverse Proportion Equations: Using Proportionality Reasoning To Find The Value Of K Or X

y = k/x

dy/dx = -k/x^2

The equation y = k/x is an inverse proportion equation. This means that as x increases, y decreases and vice versa. The value of k is a constant that determines the specific relationship between x and y.

If we graph this equation, the graph would be a hyperbola with the x-axis and y-axis as asymptotes. This is because as x approaches 0, y approaches infinity, and as x approaches infinity, y approaches 0.

To solve problems involving this equation, we can use proportionality reasoning. For example:

1. If y = 8 when x = 4, what is the value of k in the equation y = k/x?

We can use proportionality reasoning by setting up the ratio of y to x and setting it equal to the ratio of any other pair of values for y and x:

y/x = 8/4

Simplifying, we get:

y/x = 2

Now we can substitute this ratio into the equation y = k/x:

2 = k/4

Solving for k, we get:

k = 8

So the equation is y = 8/x.

2. If k = 24 and y = 6, what is the value of x in the equation y = k/x?

Again, we can use proportionality reasoning to set up a ratio of y to x and set it equal to the ratio of any other pair of values for y and x:

y/x = 6/x

Now we can substitute in the value of k and the equation y = k/x:

6/x = 24/x

Solving for x, we get:

x = 4

So the equation is y = 24/4x or y = 6/x.

In summary, the equation y = k/x represents an inverse proportion, where y decreases as x increases and vice versa. To solve problems involving this equation, we can use proportionality reasoning to set up ratios and solve for the constant k or the variable x.

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