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In algebra, one of the fundamental steps to solve equations is to rearrange them. When we talk about rearranging equations, we're essentially shifting the components of the equation to make it easier to solve for the variable in question, usually isolating it on one side of the equation.
This involves:

• Adding or subtracting terms on both sides to move variables and constants around.
• Ensuring that all terms containing the desired variable end up on one side.
• Keeping any constants, or numbers without variables, on the opposite side.

Let's consider the equation provided: \[2m - 1 = -6m + 5\]In this example, we want to gather all terms with the variable \(m\) on one side. To do that, we can add \(6m\) to both sides, allowing us to combine the variable terms on the left and isolate them: \[2m + 6m = 1 + 5\]This step is crucial because it sets the stage for the next step by simplifying the equation structure, making it more manageable.

Once the equation has been rearranged, as illustrated in the first step, simplifying it becomes essential. Simplifying an equation means combining like terms to further reduce the complexity of the equation. This involves adding or subtracting numeric terms or combining coefficients of variables.
For the equation we're exploring, after rearranging, you'll have: \[2m + 6m = 1 + 5\]Here, we simplify by:

• Combining like terms on each side: \(2m + 6m\) becomes \(8m\).
• Similarly, combining \(1 + 5\) gives \(6\).

The equation now looks much simpler: \[8m = 6\]It's easier to handle and closer to our goal of finding the value of \(m\). Simplifying is a key skill in algebra as it transforms equations into simpler ones, thus making solving them more straightforward.

Solving algebraic equations involves a systematic approach to isolate the variable and find its value. Once equations are rearranged and simplified, the final step is solving them.
To solve the simplified equation \(8m = 6\), perform the following:

• Identify the operation affecting the variable \(m\) directly, which here is multiplication by 8.
• To isolate \(m\), do the inverse operation. Divide both sides of the equation by 8.

This gives us:\[m = \frac{6}{8}\]Simplifying \(\frac{6}{8}\) by dividing both numerator and denominator by 2, you'll find:\[m = \frac{3}{4}\]Checking the solution is the last verifying step. Substitute \(m = \frac{3}{4}\) back into the original equation to ensure that both sides equal.
Each of these steps ensures that you methodically approach the solution of not just this equation, but any linear algebraic equation swiftly and accurately.