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When solving linear equations, the first step is to simplify both sides of the equation. Simplifying makes it easier to work with and to understand the relationship between the variables involved here.

Let’s take a closer look at our problem:

• Original Equation:

  $$ 3(t+2) + t = 2(3+2t) $$

• Expand the terms inside the parentheses. This means you multiply the number outside the parentheses by each term inside.
From there:
Left side:

$$3(t+2) + t = 3t + 6 + t = 4t + 6 $$


Right side:

$$2(3 + 2t) = 6 + 4t $$

After expanding, our equation becomes:

$$ 4t + 6 = 6 + 4t $$

Now both sides of the equation look simpler and it's easier to identify the next steps in solving this linear equation.

After simplifying an equation, we sometimes encounter certain scenarios that will help determine if the equation is actually true for all values of the variables involved.

In algebra, an **identity** is an equation that holds true no matter what values the variables take on. In this specific example, after simplification and removing the variable terms from both sides, we ended up with:

• $$4t + 6 = 6 + 4t$$

  Subtracting the same term (4t) from both sides simplifies the equation further, leading to

  $$6 = 6$$

• As shown, this final equation confirms the statement is true regardless of the value of 't'. Hence, $$3(t+2) + t = 2(3+2t)$$ is an identity.

This is a very useful concept since it helps identify equations that don’t depend on specific variable values but are universally valid.

To solve linear equations, we often need to isolate the variable. Isolated variables make the equation easier to manage and solve.

In practice, isolating a variable typically involves collecting like terms and performing operations that help move terms involving the variable to one side of the equation and constant terms to the other.

Example: Let's take a simplified equation form:

• $$4t + 6 = 6 + 4t$$

In this scenario, subtracting

• $$4t $$

  b from both sides yields:

• $$6 = 6$$


This subtraction step removes the variable from both sides, making it clear we have an identity.
Understanding how to isolate variables is essential for solving more complex equations where identifying such relationships promptly streamline finding solutions.