91Ó°ÊÓ

When solving linear equations, the first step often involves combining like terms. Like terms are terms that contain the same variable raised to the same power. In our exercise, the terms involving 'w' on the left side of the equation are: \(8w\), \(-4w\), and \(w\). We combine them by adding and subtracting the coefficients (the numerical part), while keeping the variable part the same. So, we perform the operations: \(8w - 4w + w\). This simplifies to \(5w\). This step reduces the complexity of the equation, making it easier to solve. Remember, combining like terms helps in reducing the number of terms, and this simplification is crucial for solving the rest of the equation.

Once we've combined the like terms, the next step is to simplify the equation further. In the exercise, we simplified the left side to get \(5w = -3\). Simplification is about making the equation easier to work with by combining like terms and reducing any complex expressions. Simplifying an equation makes it clearer and more manageable. After combining our terms, the equation became straightforward. This step is important because a simplified equation allows us to clearly see the relationship between the variables and the constants.

The final step in solving an equation is checking the solution. This is important to ensure that the solution is correct. We do this by substituting the found value back into the original equation. In our case, we determined that \(w = -\frac{3}{5}\). To check our solution, we substitute \(w = -\frac{3}{5}\) back into the original equation \(8 w-4 w+w=-3\). After substitution, we perform the arithmetic operations and simplifications: \[8(-\frac{3}{5}) - 4(-\frac{3}{5}) + (-\frac{3}{5}) = -3 \] Simplified, this confirms: \left( \frac{-24}{5} + \frac{12}{5} - \frac{3}{5} = -\frac{15}{5} = -3 \right). Since both sides of the equation match, our solution is verified as correct. This step is critical because it confirms the accuracy of our solution, ensuring that we haven't made any mistakes in our calculations.