91Ó°ÊÓ

When working with linear equations, understanding equation operations is crucial. These operations are based on the properties of equality, which state that what you do to one side of the equation, you must also do to the other. Starting with our example, the equation is set as \(0.04x - 0.2 = 0.07\). Here, you have a combination of multiplication (the \(0.04x\)) and subtraction (the \(-0.2\)).

To unravel the equation, we perform the inverse operations. Since subtraction is easier to deal with first, we add \(0.2\) to both sides, ensuring the equation remains balanced. This is an example of an equation operation that modifies the equation towards simplicity, turning it into \(0.04x = 0.27\), moving us closer to the solution.

The goal of solving an equation is to isolate the variable, in other words, to get the variable by itself on one side of the equation. After our initial equation operation, we have \(0.04x = 0.27\). To isolate \(x\), we need to eliminate the coefficient, \(0.04\), that is multiplying \(x\).

To do this, we use the inverse of multiplication, which is division. By dividing both sides of the equation by \(0.04\), we are performing an operation that will remove the coefficient completely, leaving us with the isolated variable: \(x = 6.75\). It's important to understand that every step taken must maintain the equation's balance to lead to the correct solution.

Once we have a solution for our variable, such as \(x = 6.75\), it's essential to verify that this solution is accurate. To do this, we substitute our found value back into the original equation and perform the necessary operations to see if the equation holds true. For our example, we place \(6.75\) back into the equation, which gives us \(0.04 * 6.75 - 0.2\).

Upon calculating, we find that it simplifies to \(0.07\), exactly the right side of our original equation. The left side matches the right side, confirming that our solution \(x = 6.75\) is correct. It’s an essential practice to check your solutions to avoid errors and ensure understanding of the processes used to solve the equation.