91Ó°ÊÓ

When solving linear equations, combining like terms is a crucial step. Like terms are terms that contain the same variables raised to the same power. For example, in the equation \[ \frac{1}{3} x = 2 - \frac{2}{3} x \], both sides of the equation have terms with the variable \(x\). To combine these like terms, you'll want to get all the \(x\) terms on one side of the equation. Here's how to do it for our example:
- First, add \[ \frac{2}{3} x \] to both sides of the equation: \[ \frac{1}{3} x + \frac{2}{3} x = 2 \]
- Next, combine these terms: \[ \frac{1}{3} x + \frac{2}{3} x = \frac{3}{3} x \]. Simplifying the fraction \[ \frac{3}{3} \] gives you \(1\), so the equation simplifies to \[ x = 2 \].
Combining like terms in this manner helps to simplify the equation and makes it easier to solve.

Fractions can make equations look more complicated than they are, but they can be easily eliminated. In the equation \[ \frac{1}{3} x = 2 - \frac{2}{3} x \], the presence of \[ \frac{1}{3} \] and \[ \frac{2}{3} \] can be confusing.
To eliminate these fractions, you should aim to combine terms as described earlier. Alternatively, you could also multiply every term by the denominator to clear the fractions. However, in this case:
- By adding \[ \frac{2}{3} x \] to both sides: \[ \frac{1}{3} x + \frac{2}{3} x = 2 \]
- It results in \[ \frac{3}{3} x = 2 \].
With \[ \frac{3}{3} = 1 \], this simplifies directly to \[ x = 2 \].
By efficiently combining like terms, you'll find that fractions naturally get eliminated as you simplify the equation.

Balancing equations means ensuring that both sides of an equation remain equal as you perform operations to isolate the variable. Take the equation \[ \frac{1}{3} x = 2 - \frac{2}{3} x \]:
- To balance it, you start by adding \[ \frac{2}{3} x \] to each side: \[ \frac{1}{3} x + \frac{2}{3} x = 2 \]
- Combine like terms: \[ \frac{3}{3} x = 2 \]
- Simplify: \[ x = 2 \]
At each step, we performed the same operation on both sides of the equation. This balancing act is crucial to maintaining the equation's validity. Balancing is essential because it ensures that the equality holds true, leading to the correct solution. When you substitute \[ x = 2 \] back into the original equation, both sides remain equal, confirming the solution's accuracy.