Understanding Inverse Equations
When we talk about inverse equations, we refer to a pair of equations that have opposite operations. In other words, if we apply one equation to a certain variable, applying the inverse equation will reverse the process and bring the variable back to its original value.
For instance, if we have an equation that adds 5 to a variable, the inverse equation will subtract 5 from the same variable. Similarly, if we have an equation that multiplies a variable by 2, the inverse equation will divide the same variable by 2.
Finding the Inverse Equation of y = 100×2
Now, let’s apply this concept to the equation y = 100×2. To find the inverse equation, we need to isolate x and express it in terms of y.
Starting with the given equation:
y = 100×2
We can divide both sides by 100:
y/100 = x2
To isolate x, we need to take the square root of both sides:
sqrt(y/100) = x
Therefore, the inverse equation of y = 100×2 is:
x = sqrt(y/100)
Understanding the Inverse Equation
Now that we have found the inverse equation, let’s see what it represents. The equation x = sqrt(y/100) means that if we have a certain value of y, applying this equation to it will give us the corresponding value of x that makes the original equation true.
For example, let’s say y = 400. We can plug this value into the inverse equation:
x = sqrt(400/100) = sqrt(4) = 2
Therefore, the x value that corresponds to y = 400 is 2. If we substitute x = 2 into the original equation y = 100×2, we get:
y = 100(2)2 = 400
As expected, the original equation is true for x = 2 and y = 400.
Another way to understand inverse equations is by looking at their graphical representation. The graph of y = 100×2 is a parabola that opens upwards, as shown below:
On the other hand, the graph of its inverse equation x = sqrt(y/100) is a reflection of the original graph over the line y = x:
Notice that the inverse graph is also a parabola, but it opens to the right instead of upwards. This reflects the fact that applying the inverse equation to the y value will give us two possible values of x, one positive and one negative, due to the square root operation.
Inverse equations are a powerful tool in mathematics that allow us to reverse the effect of a given equation. In the case of y = 100×2, we have found that its inverse equation is x = sqrt(y/100), which represents the value of x that makes the original equation true for a given value of y. Graphically, the inverse equation corresponds to a reflection of the original graph over the line y = x. Understanding inverse equations can help us solve problems and gain a deeper insight into the workings of mathematics.