Problem 77 - Intermediate Algebra

Question

Solve each equation. $$ 8 x=x $$

Step-by-Step Solution

Answer

The solution is \(x = 0\).

Step 1: Set the Equation

The given equation is \(8x = x\). We need to solve the equation to find the value of \(x\).

Step 2: Collect Like Terms

To isolate the variable on one side, subtract \(x\) from both sides to get like terms together. The equation becomes: \(8x - x = 0\).

Step 3: Simplify the Equation

Combine the terms on the left side: \(8x - x\) simplifies to \(7x\). So the equation is now: \(7x = 0\).

Step 4: Solve for x

Divide both sides of the equation by 7 to get \(x\) by itself: \(x = 0/7\).

Step 5: Simplify the Solution

Simplify \(x = 0/7\) to \(x = 0\).

Key Concepts

Understanding Algebra

Algebra might seem like an intimidating subject at first, but it's essentially about finding unknown values with the help of mathematical expressions. Think of algebra as a form of abstraction where letters, like \(x\), are used to stand in for numbers.
This abstraction lets us express and solve real-world problems with ease. The goal in algebra is often to solve for one or more unknown variables, commonly by performing operations to both sides of an equation.
Consider our exercise: the equation \(8x = x\). This equation implies there's some value of \(x\) that makes both sides equal. Our task is to figure out what \(x\) is.

Variable Isolation

Variable isolation is a fundamental step in solving equations. It involves getting the variable, often represented by \(x\), all by itself on one side of the equation.
This helps us clearly see what value the variable can take. In our example, starting with the equation \(8x = x\), we aim to move all the terms involving \(x\) to one side.
You do this by performing operations that will strip away the other terms, much like peeling layers away from an onion. We subtract \(x\) from both sides to begin this isolation process:
\(8x - x = 0\). At this stage, all terms with \(x\) are now on one side of the equation.

Simplification

Once the variable is isolated on one side, like with our expression \(8x - x = 0\), the next step is simplification. Simplifying often means combining like terms or performing arithmetic operations to make the equation as neat as possible.
This usually helps in getting the equation to a state where solving for the unknown becomes straightforward. In practice, \(8x - x\) simplifies to \(7x\). This means, you consolidate the terms by performing the subtraction in this case, leading to \(7x = 0\).
The simplified form \(7x = 0\) is much easier to handle when solving for \(x\) in the next steps.

Equation Solving

With our equation now simplified to \(7x = 0\), we focus on solving for \(x\). Equation solving is the culmination of our previous steps, where isolation and simplification have made finding the actual value of \(x\) feasible.
To isolate \(x\) entirely, divide both sides of the equation by 7. This yields \(x = 0/7\). Mathematically, dividing by 7 effectively cancels out the coefficient of \(x\) on one side of the equation.
We're left with \(x = 0/7\), which simplifies directly to \(x = 0\). This division and simplification mark the completion of the solution process, leaving us with the answer to the given equation.