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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