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Equations are math sentences that use an equal sign to show that two expressions on both sides are equal. Solving an equation is the process of finding the value of the variable that makes this statement true. In algebra, we often deal with linear equations, which involve variables raised to the power of one.
To solve equations, you need to perform operations that keep both sides equal while progressively simplifying the equation. In our example, we start with the equation \(18 - 3x = -24\). Here, the goal is to find the value of \(x\) by performing operations that transform this equation into a simpler form.
The basic strategy is to use the opposite operation to each term until the variable is isolated. This might involve addition, subtraction, multiplication, or division as needed. Remember, whatever you do to one side of the equation, you must do to the other to maintain balance.

Algebraic manipulation involves rearranging equations using mathematical operations to simplify them or solve for a variable. One of the core skills in algebraic manipulation is performing the same action on both sides of the equation to maintain balanced equality.
In our example, we have the equation \(18 - 3x = -24\). The first step to manipulate this equation is to move constant terms to one side and variables to the other. We do this by subtracting 18 from both sides:

• Subtract 18 from the left to eliminate the 18: \(18 - 3x - 18\).
• Do the same operation on the right: \(-24 - 18\).

This results in \(-3x = -42\). Such manipulation simplifies the steps needed to isolate the variable. Notice how each operation is mirrored on both sides of the equation to keep it balanced.

Isolating the variable means getting the variable by itself on one side of the equation. This often involves a series of algebraic modifications, especially when coefficients are attached to variables.
In our equation, \(-3x = -42\), the next step after algebraic manipulation is getting \(x\) alone. Since \(x\) is multiplied by \(-3\), you'll need to divide both sides by \(-3\) to "undo" this multiplication. This is a critical step in algebraic problem solving:

• Divide the left side: \(\frac{-3x}{-3}\) results in \(x\).
• Divide the right side: \(\frac{-42}{-3}\) simplifies to \(14\).

Completing these steps gives us \(x = 14\). The variable \(x\) is now isolated, and the original equation is solved. This process is fundamental in algebra and provides the answer we're looking for.