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When it comes to solving equations, the core task is to find the value of the variable that makes the equation true. In our example, the variable is \(z\). Solving an equation often involves several strategic steps, which could include:

• Combining like terms
• Moving terms to one side of the equation
• Using arithmetic operations such as addition, subtraction, multiplication, and division

In each step, your objective is to simplify the equation progressively until you isolate the variable. Think of it as peeling layers off an onion: with each step, you're getting closer to the core solution.

Fractions in equations can be a bit tricky, but they follow the same basic principles of solving any equation. The main challenge is often about eliminating the fraction to simplify the equation. Here are a few tips:

• Look for a common denominator between fractions on the same side. This allows you to combine them into one term.
• Multiply through by the variable, or a common denominator, to remove fractions. This helps transform the equation into a linear form, which is easier to manage.

For the exercise, adding \[\frac{16}{z} + \frac{12}{z} = -7\] combines the fractions into a single term by the common denominator \(z\). Eliminating fractions means multiplying both sides by \(z\), effectively canceling out the denominators.

Linear equations are equations of the first degree, meaning they involve no powers higher than one in any terms. The general form of a linear equation is \(ax + b = c\). Solving a linear equation involves manipulating the equation to isolate \(x\) (or the variable being solved).

In the given step-by-step solution, we turned a fraction equation into a linear equation by eliminating the denominator, leading to:\[28 = -7z\]From here, it's straightforward to solve by dividing both sides by \(-7\) to get:\[z = -4\]Linear equations are some of the simplest forms of equations you'll encounter, but mastering them is essential for understanding more complex mathematical concepts. They effectively allow for the foundational practice of algebraic manipulation and logic.