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The distributive property is a useful algebraic tool. It allows you to multiply a single term across terms inside parentheses. It states that for any numbers a, b, and c, the following is always true: \( a(b + c) = ab + ac \).

In the context of solving equations, this property helps us eliminate the parentheses so we can work with simpler expressions. Let’s take a look at the initial step:

For the given equation: \( -5(1-2z) + 4(3-z) - 7(3+z) = 0 \)

We distribute the constants as follows: \( -5(1) - 5(-2z) + 4(3) + 4(-z) - 7(3) - 7(z) \). This simplifies to: \( -5 + 10z + 12 - 4z - 21 - 7z \).

Notice how each term outside the parentheses is multiplied by each term inside? That's the distributive property in action! Applying it correctly helps clear the way for the next steps of solving the equation.

Combining like terms is an essential step in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power.

In our given equation, after applying the distributive property, we have: \( -5 + 10z + 12 - 4z - 21 - 7z \).

To combine like terms, we bring together the ones with the variable z: \( 10z - 4z - 7z \) and the constants: \( -5 + 12 - 21 \).

Adding these gives us: \( -14 - z \).

This combined expression is much simpler and easier to work with. By grouping terms with the same variable and simplifying, we reduce the complexity of the equation significantly.

Once we have simplified the equation by distributing and combining like terms, it’s time to isolate the variable. This means we want the variable by itself on one side of the equation.

From our previous steps, we have: \( -14 - z = 0 \).

To isolate z, we need to get rid of the -14. We do this by adding 14 to both sides: \( -z = 14 \).

Now, to solve for z, we multiply both sides by -1: \( z = -14 \).

By isolating the variable, we make it easy to see the solution to the equation. It’s important to follow these steps systematically to ensure there are no mistakes. That's it! We've solved for z.