91Ó°ÊÓ

When dealing with equations, our goal is to find the value of the variable that makes the equation true. Equations are like balance scales. Every operation performed on one side of the equation must also be performed on the other side to keep the scale balanced.
Begin by identifying where the variable is located within the equation. The example equation, \(3x + 54 = 360\), shows us that the variable term, \(3x\), is on the left-hand side of the equation. To solve for \(x\), we need to make sure that \(x\) stands alone on one side of the equation.

• Perform operations systematically to isolate the variable.
• Always ensure that the equation remains balanced through each step.

Remember, the key to solving equations is performing inverse operations—opposite actions that undo each other—to simplify the expression as we move towards isolating the variable.

Isolating the variable means getting the variable on one side of the equation by itself. In the equation \(3x + 54 = 360\), you first need to isolate \(3x\).
This is done by eliminating the constant term, \(54\), from the side where the variable term resides. Subtract 54 from both sides to keep the balance:
\[3x + 54 - 54 = 360 - 54\]
Simplifies to:
\[3x = 306\]

• Eliminate constants from the side with the variable by using inverse operations, such as addition and subtraction.
• Perform the same operation on both sides to maintain equilibrium.

Once isolated, the variable term can be further tackled to find the value of the variable itself by addressing any attached coefficients.

Simplifying an equation involves performing operations that make it easier to handle, ultimately leading to the solution. In our example, after isolating the variable term \(3x\), the equation becomes \(3x = 306\).

Now, we need to get rid of the coefficient, which in this case is 3. The simplest way to do this is to divide both sides of the equation by the coefficient to solve for \(x\):
\[x = \frac{306}{3}\]
After computing the division, the result is:
\[x = 102\]

• Division works as the inverse operation to multiplication when dealing with coefficients.
• Ensure that equations are simplified to their most basic form by removing coefficients.

Simplification is a crucial step that sometimes overlaps with isolating the variable, as these processes often go hand in hand, breaking down complex problems into understandable solutions.