91Ó°ÊÓ

When working with linear equations, combining like terms is a vital step to simplify expressions before solving them. Like terms in an equation are terms that have the same variable raised to the same power. For example, in terms like \(2x\), \(3x\), and \(4x\), the variable part \(x\) is the same, which allows you to combine them.

To combine like terms, simply add or subtract the coefficients. In the original exercise, the coefficients are 2, 3, and 4. Adding these together, we get \((2+3+4)x = 9x\).

This reduces the equation significantly, making it much easier to work with. Remember, combining like terms doesn't change the balance of the equation; it just simplifies it for further operations.

Once you have simplified your equation by combining like terms, the next step is to solve for the variable \(x\). Solving for \(x\) means isolating \(x\) on one side of the equation to find its value.

In our case, after combining like terms, the simplified equation is \(9x = 180\). The goal is to get \(x\) by itself. To do this, divide both sides of the equation by the coefficient of \(x\), which is 9. This operation gives us:

\[x = \frac{180}{9}\]

Solving this division gives you \(x = 20\). By performing equal operations on both sides of the equation, you maintain its balance. This method ensures that your solution is correct.

Simplifying equations is a crucial step in solving them, as it involves reducing the equation to its simplest form by performing basic arithmetic operations. A simplified equation is much easier to read and work with, reducing the likelihood of mistakes in subsequent steps.

Let’s recap the process using the original equation: \(2x + 3x + 4x = 180\). By combining like terms, we simplify this to \(9x = 180\), a much leaner equation.

From there, simplifying further involves solving for \(x\), as discussed earlier. After dividing both sides by 9, we get \(x = 20\). This result is the simplest form of the equation, and the solution verifies that \(x\) satisfies the original equation. Practice these techniques regularly to strengthen your problem-solving skills!