91Ó°ÊÓ

Solving equations is one of the foundational skills in algebra. It involves finding the values of variables that make an equation true. When we solve an equation, we are trying to determine what numbers can replace the variables to balance both sides of the equation. This requires understanding and applying various mathematical operations such as addition, subtraction, multiplication, or division.
To solve the equation, a standard approach involves performing the same operations on both sides of the equation. This keeps the two sides balanced and allows us to isolate the variable eventually. In our example, we started with the equation:

• \(-6 = \frac{k}{8}\)

Here, our goal was to solve for the variable \(k\). Multiplying both sides by 8 balanced the equation and eliminated the fraction, leading us to find \(k = -48\). Ultimately, solving equations allows us to unveil the relationships between quantities expressed through variables.

Isolating a variable is a technique used in algebra to simplify equations so that the variable stands alone on one side. The purpose is to make it easier to understand what values the variable can take. To isolate a variable means to perform all necessary operations to "unlock" the variable from the other numbers and symbols in the equation.
It often involves reversing the operations that have been applied to the variable, such as division or multiplication. In our exercise, the variable \(k\) was inside a fraction. To isolate \(k\), we had to multiply both sides by 8, which effectively cancels the `/8`, leaving us with \(k\) on one side. This was illustrated in the step:

• \((-6)(8) = \frac{k}{8}(8)\)

In general, isolation of variables is a crucial step in solving any algebraic equation, as it clears the path to find the solution.

Dealing with fractions in equations can initially seem tricky, but with some practice, it becomes much more manageable. Fractions are simply divisions expressed differently: the numerator is divided by the denominator.
When equations involve fractions, one effective method to clear the fractions is to multiply every term by the least common denominator (LCD). This step removes the fractions and simplifies the equation. In our example, the equation had:

• \(-6 = \frac{k}{8}\)

By multiplying both sides by 8, we eliminated the fraction \(\frac{k}{8}\) to simplify to \(-48 = k\).
Clearing fractions early in the solution process makes it more straightforward to perform operations like addition or subtraction needed for solving the equation. This process not only simplifies the equation but can help avoid common mistakes, making solving the entire equation much more intuitive.