91Ó°ÊÓ

To solve an equation, one of the first and most crucial steps is to isolate the variable. This means you want to get the variable, often represented by a letter like \(x\), alone on one side of the equation. By doing this, you effectively untangle the equation, making it easier to see what the variable equals. In the given exercise, the equation is \(-x + 8 = 13\). The goal is to get \(x\) all by itself on one side.
Start by removing any constants or numbers that are on the same side as \(x\). Here, subtracting \(8\) from both sides helps strip away anything extra surrounding \(x\):

• Subtract \(8\) from both sides: \(-x + 8 - 8 = 13 - 8\).
• You'll simplify to: \(-x = 5\).

After this step, \(x\) is not yet isolated because of the negative sign. However, you have eliminated the constant, which is a vital part of isolating \(x\). Doing this prepares you for solving the equation fully.

Linear equations are equations where the variable is raised to the power of one. These equations form a straight line when graphed, hence their name. Solving them involves finding the value of the variable that makes the equation true. In our example, we have the equation \(-x = 5\).
Since the variable \(x\) is already isolated barring the negative sign, the task becomes to solve for \(x\) by addressing this negative:

• The next step is to eliminate the negative sign by multiplying both sides by \(-1\).
• This changes the equation from \(-x = 5\) to \(x = -5\).

By completing this step, you find that \(x = -5\), which is the solution to the equation. This value, when plugged back into the original equation, maintains the balance and hence verifies the solution.

Inverse operations are operations that undo each other, like addition and subtraction or multiplication and division. They are essential in solving equations because they help remove any operations applied to the variable. For example, if you add a number to the variable, you can use subtraction to eliminate it. Similarly, multiplication can be undone by division.
The exercise uses subtraction and multiplication, both of which have inverse operations:

• To remove the \(+8\) alongside \(-x\), subtract \(8\), as subtraction undoes addition.
• To handle the \(-1\) coefficient on \(x\), multiply both sides by \(-1\), cancelling out the negative sign.

This strategic use of inverse operations simplifies the equation until the variable \(x\) is isolated and solved. Recognizing and applying these operations is key to efficiently solving equations of this type.