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Chapter 2: Problem 10

Solve each equation. $$ 2 x=x-5 $$

Short Answer

Expert verified
\(x = -5\).

Step by step solution

01

Combine like terms

Start by subtracting \(x\) from both sides of the equation to move all the \(x\) terms to the left side. This gives us: \(2x - x = -5\).
02

Simplify the equation

Combine the \(x\) terms on the left side. \(2x - x\) simplifies to \(x\). Now, the equation becomes \(x = -5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms
When solving linear equations, the first step is often to bring similar terms together. This is known as "combining like terms." In our example, the equation is \(2x = x - 5\). The first action is to consolidate all terms with \(x\) to one side of the equation.
  • The terms "like" each other if they have the same variable raised to the same power. In this case, both terms are \(x\) terms, so they are like terms.
  • We move the \(x\) from the right side to the left side by subtracting \(x\) from both sides, resulting in \(2x - x = -5\).
This process simplifies the equation and prepares it for the next stages of solving.
Simplifying Equations
Simplifying equations is the process of making the equation as straightforward as possible. We do this by performing operations that reduce the equation to a simpler form. In the example equation, we've already combined like terms to get \(2x - x = -5\).
  • The term \(2x - x\) simplifies further because \(2x - x = x\). We just have to subtract the coefficients (numbers in front of the \(x\)) to get \(x\).
  • This simplification converts the equation into \(x = -5\), a much simpler form that is easy to understand and analyze.
This new form of the equation makes it easier to see the solution.
Subtracting Terms
Subtracting terms is a critical step when you are moving terms from one side of an equation to the other. This action helps isolate the variable on one side, which simplifies the equation. For our problem, we started with \(2x = x - 5\).
  • To bring all \(x\) terms to the left, we subtract \(x\) from both sides. This means we perform the operation \(2x - x\) on the left side, resulting in \(x\).
  • Doing the same operation on both sides keeps the equation balanced and ensures equality.
After subtracting, it allows the equation to transition from a more complex form to a much simpler one, like \(x = -5\), making it easy to interpret and solve for the variable.

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