91Ó°ÊÓ

Algebraic manipulation is the foundational process of rearranging and simplifying equations to solve them. We use operations like addition, subtraction, multiplication, and division while respecting algebraic rules. Our aim is often to either

• combine like terms,
• move terms from one side to another, or
• simplify expressions for easier interpretation.

In our example equation, we start by performing algebraic manipulation to group all terms that contain the variable \(w\) on one side. To do this, we subtract \(3w\) from both sides. By following this approach, we create a clearer path towards solving the equation.
This aligns all variable terms allowing for a direct path to solution finding. If the variable terms were left on both sides, it becomes difficult to isolate and solve the variable being targeted.

Once the algebraic manipulation has regrouped our terms, the next step is variable isolation. This concept involves getting the variable alone on one side of the equation to identify its value. Doing so generally requires reversing operations that impact the variable.
In our exercise, after we simplify to \(7 = 4 + 6w\), we isolate \(w\) by removing any constants from the side containing the variable. We accomplish this by subtracting 4 from both sides, resulting in a cleaner equation: \(3 = 6w\).
This step brings us closer to resolving what \(w\) should equal, carving out an uncomplicated space for calculation. Isolating a variable might involve multiple operations depending on the complexity of the equation, but systematic reversal of earlier operations will ensure we maintain equation balance.

Simplification is reducing an expression into its most basic form. It's necessary in almost every stage of equation solving, as it makes handling and interpreting results much easier.
In the example process, after isolating \(w\), the equation \(3 = 6w\) needs simplification to reveal the value of \(w\). We do this by dividing both sides by 6, which neutralizes the multiplier effect on \(w\). This division gives us \(w = \frac{3}{6}\).
The last step of simplification transforms \(\frac{3}{6}\) into \(\frac{1}{2}\), showing \(w\)’s equivalency in its simplest form. This final simplification step ensures the solution is presented in the cleanest and most understandable way possible for any further mathematical or practical application.