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An equation is like a balance scale, where two expressions are equal to each other. It consists of two sides separated by an equal sign (=). Each side of an equation can contain numbers, variables, and mathematical operations. An important thing to remember is that whatever you do to one side of the equation, you must do to the other side, in order to keep the balance.

To solve an equation, the goal is to find the value of the variable that makes both sides of the equation equal. This is done through a series of steps, which includes operations such as addition, subtraction, multiplication, or division. In the given exercise, the equation is:

• Left side: 4x + 7
• Right side: 5 - 8x

Here, the variable is \(x\), which you will solve for by isolating it on one side of the equation.

Variable isolation is a crucial step in solving equations. It involves rearranging the equation so that the variable stands alone on one side, usually the left side. This is done to clearly see what the variable equals.

In the problem at hand, the first step of variable isolation is moving all the terms in the equation that contain \(x\) to one side. You achieve this by manipulating both sides of the equation equally. For example, to move -8x from the right side to the left, you add 8x to both sides:

• Initially: \(4x + 7 = 5 - 8x\)
• Add 8x to both sides: \(4x + 8x + 7 = 5\)
• Simplify: \(12x + 7 = 5\)

Now, you have an equation where \(x\)-terms are gathered on one side, making it easier to proceed towards isolating \(x\).

Solving linear equations is the process of finding the unknown variable that makes an equation true. These equations are 'linear' because the variable is not raised to any power other than one.

The steps in the exercise involve solving the linear equation \(12x + 7 = 5\). To isolate \(x\), you first need to deal with the constant number 7 by subtracting it from both sides:

• Subtract 7: \(12x + 7 - 7 = 5 - 7\)
• Which simplifies to: \(12x = -2\)

Once the terms that do not involve \(x\) are removed, you have \(12x = -2\). To find \(x\), divide both sides by 12:

• \(x = \frac{-2}{12}\)
• Simplify: \(x = \frac{-1}{6}\)

Thus, the solution \(x = \frac{-1}{6}\) is the value of the variable that balances the equation.