91Ó°ÊÓ

When it comes to solving linear equations, the distributive property is a useful tool. In essence, this property allows us to multiply a single term by each term inside a parenthesis. For example, if we have an equation such as \(1 x=3(x+2)\), we apply the distributive property in the following manner: multiply the number outside the parenthesis, which in this case is 3, by each term inside the parenthesis \(x+2\). So, it would look like \(3*x + 3*2\), simplifying to \(3x + 6\).

Understanding and correctly applying the distributive property is crucial to simplify equations and combine like terms, which sets the stage for further steps in solving the equation.

Rearranging equations is a fundamental skill in algebra that involves moving terms from one side of the equation to the other. This often follows the use of the distributive property. In the given problem, after applying the distributive property, we get \(3x + 6\). In the next step, we need to gather like terms to simplify the equation.

To do that, we want to bring all the variable terms (involving 'x') to one side and the constants to another. From our example, we subtract \(1x\) from both sides which leads to \(3x - 1x\), simplifying to \(2x\) on one side of the equation. The constant \(6\) moves to the other side of the equation by subtraction, resulting in \(2x =-6\). This is a prime example of rearranging the equation to pave the way for isolating the variable 'x'.

Isolating the variable is the final step in solving linear equations. After rearranging the terms, we often end up with the variable term on one side and a constant on the other, as seen with \(2x = -6\). To find the value of 'x', we need to isolate it by performing the opposite operation of what is currently being applied to the variable. In the provided exercise, 'x' is multiplied by 2, so we'll do the opposite: divide both sides by 2. This operation gives us \(\frac{-6}{2}\) or \(x = -3\), successfully isolating 'x' and solving the equation.

It's important to treat both sides of the equation equally in this step; whatever you do to one side, you must do to the other to maintain the balance of the equation. Understanding how to isolate the variable is essential in finding the solution to the problem.