91Ó°ÊÓ

When solving linear equations, the first step is often to combine like terms. In our example, the equation given is \(9x - 3x + x = -4\). Like terms are the terms that contain the same variable. Here, we have three terms involving x: \(9x\), \(-3x\), and \(x\). To combine them, add or subtract the coefficients of these like terms. This step simplifies the equation and makes it easier to solve. In our case, \(9 - 3 + 1 = 7\). Hence, combining these like terms gives us \(7x = -4\). Combining like terms helps to reduce the equation to a simpler form that is easier to handle in the subsequent steps.

Once we have combined like terms, the next step is to isolate the variable, which in this case is \(x\). This means we need to get \(x\) by itself on one side of the equation. From our simplified equation \(7x = -4\), we can see that \(x\) is being multiplied by 7. To isolate \(x\), we need to undo this multiplication, which we do by performing the opposite operation—division. Divide both sides of the equation by 7. Hence, we get \(x = \frac{-4}{7}\). This means we have successfully isolated \(x\), finding that \(x = -\frac{4}{7}\). Isolating the variable is crucial for identifying the value of the variable that satisfies the equation.

Verifying our solutions is an essential step in solving linear equations. This ensures that our solution is correct. To verify, substitute the value you found for the variable back into the original equation. For our exercise, we need to substitute \(x = -\frac{4}{7}\) back into \(9x - 3x + x = -4\). Let’s do the math: \(9 \times \frac{-4}{7} - 3 \times \frac{-4}{7} + \frac{-4}{7} = -4\). Simplifying each term, we get \(-\frac{36}{7} + \frac{12}{7} - \frac{4}{7}\). Adding these terms together, we ultimately get \(-4\), which matches the right-hand side of the original equation. Since the left side equals the right side after substituting, this confirms our solution is correct. Verification helps to catch any mistakes and confirms the reliability of our solution.