91Ó°ÊÓ

Combining like terms is an essential step in simplifying linear equations. It helps to make the equation more manageable. In our example, the terms involving the variable r (5r, -3r, and 9r) are combined.
First, you identify and group the terms with the same variable. Here, it's the r terms:

\[ 5r - 3r + 9r \]
Next, add or subtract the coefficients (the numbers in front of the variable) accordingly. For our terms, that would be:

\[ (5 - 3 + 9)r = 11r \]
Now the equation is much simpler:

\[ 11r = 33 \]
This process makes it easier to isolate the variable in the next step. Remember, combining like terms is like combining apples with apples and oranges with oranges; you can only combine similar items.

Isolating the variable is the process where you get the variable by itself on one side of the equation. In our case, the variable is r. Starting from:

\[ 11r = 33 \]
You need to divide both sides of the equation by the coefficient of r, which is 11. This step looks like this:

\[ r = \frac{33}{11} \]
This simplifies to:

\[ r = 3 \]
The goal is to have the variable alone on one side, which lets you find its value directly. It’s like peeling away the layers until you get to the core. This step can involve different operations like adding, subtracting, multiplying, or dividing, depending on the equation.

Verifying solutions ensures that the value you found for the variable is correct. To do this, substitute the variable back into the original equation and check if both sides of the equation are equal.
For our example, substitute \( r = 3 \) back into the original equation:

\[ 5(3) - 3(3) + 9(3) = 35 - 2 \]
Calculate each term:

• 5(3) = 15
• -3(3) = -9
• 9(3) = 27

Add them together:

\[ 15 - 9 + 27 = 33 \]
On the right-hand side, you have:

\[ 35 - 2 = 33 \]
Since both sides of the equation are equal (33 = 33), your solution \( r = 3 \) is verified and correct.
Always take this step to ensure accuracy. It's like checking your work to confirm that the solution you derived fits perfectly within the original problem.