91Ó°ÊÓ

One of the first steps in solving equations is to 'isolate the variable'. This means rearranging the equation so that the variable you are solving for is on one side of the equation, and everything else is on the other side.
To isolate a variable, you generally use inverse operations to move other terms or coefficients to the opposite side.
For example, in the exercise given, we aim to isolate the variable \(d\) starting with the equation \(I=\frac{ks}{d^2}\).
To isolate \(d^2\), the first step is to eliminate the fraction. Multiplying both sides by \(d^2\) gives us \(I \times d^2 = ks\). Now, \(d^2\) is no longer in the denominator and is instead part of a product on the left side of the equation.

Inverse operations are mathematical operations that undo each other. Common pairs of inverse operations include addition and subtraction, as well as multiplication and division.
In the supplied exercise, we use inverse operations to isolate \(d^2\).
Initially, we have \(I \times d^2 = ks\). To isolate \(d^2\), we divide both sides by \(I\). This is because division is the inverse operation of multiplication. So dividing both sides by \(I\) cancels out the multiplication with \(d^2\) on the left side:

• \(I \times d^2 \rightarrow d^2 = \frac{ks}{I}\)

By following these steps, we've isolated \(d^2\) using inverse operations.
This highlights the importance of understanding and applying inverse operations to manipulate algebraic expressions effectively.

Taking the square root is an important operation when working with equations that include squared terms. The square root of a number \(y\) is a value \(x\) such that \(x^2 = y\).
In our exercise, once we have isolated \(d^2\) as \(d^2 = \frac{ks}{I}\), the final step is to solve for \(d\) itself by taking the square root of both sides:

• \(d = \sqrt{\frac{ks}{I}}\)

It's essential to remember that taking the square root gives two possible values, one positive and one negative. However, the context of the problem or additional information usually helps in determining the relevant solution.
In conclusion, understanding and applying the concept of the square root is crucial for solving equations involving squared terms.