91Ó°ÊÓ

When solving equations, the objective is to find the value of the variable that makes the equation true. Equations may involve constants, coefficients, and a variable, and they often require several steps to solve. The key goal is to transform the equation in such a way that the variable stands alone on one side of the equal sign.

To achieve this,

• identify any constants and coefficients associated with the variable.
• Use inverse operations to cancel them out.

This requires balancing the equation, meaning whatever operation you do to one side, you must do to the other.

For example, in the equation \[ 49 = -5 - 14x \], we start by eliminating the constant term by adding 5 to both sides, transforming the equation to \[ 54 = -14x \].At this stage, you have made a clear path to isolate and solve for the variable.

Isolating the variable is a crucial step in solving any equation. This step requires getting the variable term alone on one side of the equation, free from any other numbers or terms.

To isolate a variable, follow these steps:

• Identify the variable you want to isolate.
• Use inverse operations to move other terms (constants or other variable terms) to the opposite side of the equation.

In our example with the equation \[ 54 = -14x \], we want to isolate \( x \).
Since \(-14x\) is already alone on the right side, you just need to divide both sides of the equation by \(-14\) to complete the isolation process: \[ x = \frac{54}{-14} \].
This formula expresses the variable x without any other terms involved, simplifying the process of solving the problem.

Fractions often appear in equations. Knowing how to manipulate them is essential for solving equations effectively. A fraction consists of a numerator (top number) and a denominator (bottom number), symbolizing a division operation.

When solving equations, you might need to:

• Convert improper fractions into proper ones or decimals for clarity.
• Simplify fractions by dividing both numerator and denominator by their greatest common divisor (GCD).

For example, after solving for \( x \) in our problem, the fraction \[ x = \frac{54}{-14} \] is not fully simplified.
We simplify this by dividing both 54 and -14 by 2 (the GCD in this case), resulting in:\[ x = \frac{27}{-7} \].
Fully simplified fractions make interpreting the solution easier and more straightforward, giving a clearer insight into the equation's result.