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To solve a linear equation, the goal is to find the value of the variable that makes the equation true. This typically involves isolating the variable by itself on one side of the equation. First, recognize what operations are being performed on the variable. Next, use inverse operations to undo these steps.
For instance, in the equation \(-8z = 120\), the variable z is being multiplied by -8. To isolate z, we perform the inverse operation, which is division. We divide both sides of the equation by -8:
\[\frac{-8z}{-8} = \frac{120}{-8}\]
This simplifies to:
\(z = -15\)
By isolating the variable in this manner, we simplify the equation and determine that z equals -15. This technique is central to solving linear equations and helps in finding the precise value of the unknown variable.

Verifying your solution is a crucial step in solving equations. After isolating the variable and finding its value, it's important to substitute this value back into the original equation to ensure that it satisfies the equation.
In our example, the solution was \(z = -15\). To verify, we substitute \(z = -15\) back into the original equation:
\(-8 \times -15 = 120\)
Simplifying the left-hand side gives:
\(120 = 120\)
Since both sides of the equation are equal, our solution is verified. This step ensures that our method was correct and the value found truly solves the equation. Always remember to check your solution to avoid any mistake.

Integer division is frequently used in solving linear equations, especially when isolating a variable. In these cases, the division must be exact since integer division involves dividing one integer by another, resulting in another integer.
In our example, after we isolated the variable z by dividing both sides by -8, we performed the calculation:

• \( \frac{120}{-8} \)

This results in -15 because \(-120 / -8 = -15\). It's important to be precise with integer division as any mistakes here can lead to an incorrect solution.
Practicing integer division without a calculator can be helpful. It strengthens basic arithmetic skills and ensures you are comfortable solving linear equations manually. Remember, careful integer division is vital to solving and verifying linear equations accurately.