91Ó°ÊÓ

Isolating variables is one of the most fundamental strategies when solving linear equations. The main goal is to get the variable you are solving for on its own, usually on one side of the equation. This allows us to find its value directly. Consider the original equation given: \(-8y - 2 = 18\). To isolate the variable \(y\), we first need to deal with the constant term on the same side as \(-8y\). Since there is a \(-2\) on the left, we do the opposite operation to cancel it out: by adding \(2\) to both sides. This derives from the principle that an equation remains balanced if you perform the same operation on both sides. Once you add \(2\) to both sides, the constant cancels out, resulting in:\(-8y = 20\).Now the variable term is isolated on one side, making it much easier to solve.

Simplifying fractions is an essential part of presenting mathematical solutions in their simplest form. After isolating the variable in the equation \(-8y = 20\), we divide by \(-8\) to solve for \(y\), giving us:\(y = \frac{20}{-8}\).Fractions are typically simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this equation, the GCD of 20 and 8 is 4. Thus, we divide both the numerator and the denominator by 4:

• 20 ÷ 4 = 5
• -8 ÷ 4 = -2

This gives us the simplified fraction:\(y = -\frac{5}{2}\).Simplifying fractions not only makes numbers easier to understand but also provides a cleaner solution.

The process for solving linear equations such as \(-8y - 2 = 18\) involves a strategic series of steps designed to simplify and solve the equation efficiently. Here's a straightforward outline of these steps:

• Step 1: Simplify Both Sides
  Check for simple arithmetic that can be simplified on either side of the equation to make things clearer.
• Step 2: Isolate the Variable
  Use inverse operations to shift all terms with the variable to one side, and constants to the other. We did this by adding 2 to both sides to cancel out the constant term on the side with \(-8y\).
• Step 3: Solve for the Variable
  Once the variable term is isolated, use division or multiplication to solve for the variable. This often involves simplifying fractions, as we did by dividing to find \(y = -\frac{5}{2}\).
• Step 4: Verify Solutions
  Substitute the solution back into the original equation to ensure that it satisfies the equation, confirming that no errors were made during the solving process.

Following these steps carefully ensures that you can solve any linear equation reliably and correctly.