91Ó°ÊÓ

In algebra, isolating the variable means getting the variable you are interested in by itself on one side of the equation. This is crucial for solving for a specific variable. Let's take it step-by-step using the given example. Firstly, identify the variable you need to isolate—in this case, 'y'. To isolate 'y', you need to perform operations that help you move other terms away from 'y' and onto the other side of the equation.
For our exercise, we start with: \(7x - 6y = 7\). The first step is to get the term with 'y' by itself on one side. So, we add \(6y\) to both sides of the equation: \( 7x - 6y + 6y = 7 + 6y \). This simplifies to: \( 7x = 7 + 6y \). Now 'y' is in a term by itself on one side, completing the isolation process.
This technique is called isolating the variable, and it is often the first step in solving an equation for a specific variable.

Once we have isolated our variable term, the next step is algebraic manipulation. Algebraic manipulation involves performing operations equally on both sides of the equation to maintain its balance and solve for the variable. In our exercise, after isolating \(6y\), we have: \(7x = 7 + 6y\). The goal now is to solve for 'y'.
Start by getting rid of constants on the same side as 'y'. Here, we subtract 7 from both sides: \(7x - 7 = 6y\). This leaves us with the term containing 'y' simplified.
Finally, to completely solve for 'y', we must isolate 'y' from its coefficient (which is 6 in this case). To do that, we divide both sides by 6: \( \frac{7x - 7}{6} = y \). Now we've successfully manipulated the algebraic equation to solve for the variable 'y'. Remember, maintaining the balance of the equation by performing the same operations on both sides is key in algebraic manipulation.

Equation rearrangement is the process of reordering terms and simplifying expressions to solve for a particular variable. This often involves moving terms from one side of the equation to the other and simplifying. It's crucial to follow the rules of equality: whatever operation you do to one side, you must do to the other.
In our exercise: \(7x - 6y = 7\), we first added \(6y\) to both sides (\(7x - 6y + 6y = 7 + 6y\)), resulting in \(7x = 7 + 6y\). This rearranged 'y' on one side. Then we subtracted 7 from both sides to more directly isolate the term with 'y' (\(7x - 7 = 6y\)). Finally, we divided each term by 6 to solve for 'y': \( \frac{7x - 7}{6} = y \).
These steps showcase how rearranging an equation, through adding, subtracting, and dividing terms, can simplify an expression and solve for the desired variable. By rearranging equations, you create a clearer pathway to isolate and solve for specific variables.