Problem 74 - Prealgebra

Question

Solve each equation. $$x+27+54=180$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 99 \).

Step 1: Simplify the Equation

Start by simplifying the expression on the left side of the equation. Combine the constants 27 and 54.\( x + 27 + 54 = 180 \)This becomes:\( x + 81 = 180 \)

Step 2: Isolate the Variable

Subtract 81 from both sides of the equation to isolate the variable \(x\).\( x + 81 - 81 = 180 - 81 \)This simplifies to:\( x = 99 \)

Key Concepts

Equation Solving

Equations are a fundamental part of mathematics that involve finding the value of unknown variables. At its core, solving an equation is about maintaining a balance between both sides of the equation. The goal is to find a value for the variable that makes the equation true. In our problem, the equation is:

• \( x + 27 + 54 = 180 \)

This type of equation is linear, meaning the variable \( x \) is not raised to any power other than one. Linear equations are commonly seen in algebra and are usually straightforward to solve, following systematic methods. Understanding how to manipulate these equations by adding, subtracting, multiplying, or dividing both sides equally allows us to find solutions effectively.

Variable Isolation

The process of isolating a variable involves rearranging the equation so that the variable appears by itself on one side of the equality sign. To do this, you perform inverse operations to "undo" the operations impacting the variable. In our example, after simplifying, the equation becomes:

• \( x + 81 = 180 \)

To isolate \( x \), we need to get rid of the 81 that is added to it. This is done by performing the inverse operation of addition, which is subtraction. Therefore, we subtract 81 from both sides to maintain equality. This gives us:

• \( x = 180 - 81 \)

Thus, \( x \) is isolated, and we find that \( x = 99 \). Isolating the variable is an essential step in solving any equation because it reveals the value of the unknown.

Step-by-Step Solution

Following a step-by-step approach when solving equations helps in understanding and avoiding mistakes. Here's why each step is crucial:

• Simplifying: This initial step involves combining like terms and simplifying expressions on each side of the equation. It makes the equation easier to handle.In our case, \( x + 27 + 54 \) simplifies to \( x + 81 \).
• Isolating the Variable: Once simplified, this crucial step involves getting the variable all by itself on one side of the equation, using inverse operations.Here, we isolated \( x \) by subtracting 81 from both sides.
• Finding the Solution: Once the variable is isolated, you solve by performing the arithmetic operation left.Here, we found that \( x = 99 \), which makes the original equation true.

These steps provide a clear structure to tackle equations, ensuring the process is logical and grounded. It also sets a strong foundation for tackling more complex algebraic problems in the future.