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One important step when solving equations is combining like terms. This involves simplifying our expressions by merging terms with the same variable or constants.

In our example, the equation is initially written as \(7w - 9 = 13 - 4w\). The goal is to both gather similar terms together and simplify any complexity.

To do this, we'll need to move all the terms containing the variable \(w\) to one side of the equation and all the constant numbers to the other.

• Add \(4w\) to both sides to combine all \(w\) terms: \(7w + 4w = 13 + 9\).
• This simplifies the equation to \(11w = 22\).

This process of aligning terms makes the solving process straightforward, setting a clear path for the next steps.

Linear equations, like the one in our exercise, involve variables to the first power, and graph as straight lines if plotted on a coordinate system. They take the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.

In our given equation \(7w - 9 = 13 - 4w\), you can see that both terms involving \(w\) are linear due to the fact they only involve \(w\)^1, not squared, cubed, or any higher power.

Understanding that linear equations produce a straight line can also be useful when graphing. It means any solution to \(w\) will reflect where the line would intersect the x-axis in a graphed setting.

Thus, appreciating the linearity of the equation simplifies predicting outcomes and assessing whether your answer might look reasonable.

Solving equations, especially when just starting out, may feel complex. However, following a step-by-step solution provides clarity and structure.

Let's use the example we have:

• **Step 1:** Combine the terms by adding like variables and numbers together. For example, gathering all \(w\)s on one side as we saw with \(7w + 4w\) and putting constants together by computing \(13 + 9\).
• **Step 2:** Simplify the equation. By performing the calculations from the first step, you get \(11w = 22\).
• **Step 3:** Solve for \(w\) by isolating the variable. Here, divide both sides by 11, resulting in \(w = \frac{22}{11}\).

By following these organized steps, you ensure that each operation is methodologically addressed, thus avoiding common mistakes and confirming your math work is accurate.