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In algebra, isolating the variable is a crucial step for solving equations. This means arranging the equation so the variable you're solving for is by itself on one side.
Let's look at our example: \(8t + 6 = 7t + 6\). To isolate \(t\), we first subtract \(7t\) from both sides: \[ 8t + 6 - 7t = 7t + 6 - 7t \]
This simplifies to \( t + 6 = 6 \). Now \(t\) is almost isolated. We need to subtract 6 from both sides to completely isolate \(t\): \( t + 6 - 6 = 6 - 6 \). The result is \( t = 0 \).
By isolating the variable, we transform a complex equation into a simple one, making it easier to find the solution.

Verifying your solution is essential to ensure accuracy. This involves substituting the solution back into the original equation to see if it holds true.
In our example, after solving for \(t\) and finding \(t = 0\), we substitute \(0\) back into \(8t + 6 = 7t + 6\).
Substituting gives us: \[ 8(0) + 6 = 7(0) + 6 \]. Simplifying both sides, we get \( 6 = 6 \).
Since both sides of the equation are equal, our solution \(t = 0\) is correct. Equation verification ensures that the steps we took to isolate the variable did not violate any algebraic principles and that our solution is indeed valid.

Simplifying equations involves combining like terms and performing arithmetic operations to make an equation more manageable. This is a key part of solving equations effectively.
In our example, \(8t + 6 = 7t + 6\), we first simplified by subtracting \(7t\) from both sides: \[ 8t + 6 - 7t = 7t + 6 - 7t \]. This resulted in a simplified equation: \( t + 6 = 6 \).
Next, we simplified further by subtracting 6 from both sides: \( t + 6 - 6 = 6 - 6 \), leaving us with \( t = 0 \).
Each simplification step reduces the equation to more basic terms, making it easier to solve. Always perform operations that simplify your equation and make sure you do the same operation on both sides to maintain equality.