91Ó°ÊÓ

When solving equations with variables on both sides, the goal is to group all the variable terms on one side and the constant terms on the other. This makes it easier to isolate the variable and find its value. To illustrate, let's start with the equation from our exercise:

\[ 19z = 18z - 7 \]
Notice that both sides of the equation have the variable 'z'. The first step is to move the variable terms to one side. This is done by performing the same operation on both sides. Here, we subtract 18z from both sides:

\[ 19z - 18z = 18z - 7 - 18z \]
Simplifying this, we get:

\[ z = -7 \]
Now, the equation is much simpler, and we have isolated the variable.

Collecting like terms is a fundamental step in simplifying and solving equations. Like terms are terms that have the same variable raised to the same power. In the context of our exercise, we focus on collecting the 'z' terms.

Initially, we have:

\[ 19z = 18z - 7 \]
Here, 19z and 18z are like terms because they both contain the variable 'z'. To simplify, we need to collect or combine them on one side of the equation. We subtract 18z from both sides:

\[ 19z - 18z = 18z - 7 - 18z \]
The equation simplifies to:

\[ z = -7 \]
By collecting like terms, we've simplified the equation and isolated the variable 'z' on one side. This step is crucial because it sets up the equation for easy solving.

After finding a solution, it's important to verify that it is correct. This ensures you didn't make any mistakes during the solving process. Verification is done by substituting the solution back into the original equation and checking if both sides are equal.

Let's verify our solution from the exercise where we found \( z = -7 \):
Substitute \( z = -7 \) into the original equation:

\[ 19(-7) = 18(-7) - 7 \]
Calculate each side:

\[ -133 = -126 - 7 \]
Further simplifying the right-hand side, we get:

\[ -133 = -133 \]
Both sides are equal, which confirms that \( z = -7 \) is indeed the correct solution. Verifying solutions helps build confidence that your answer is accurate.