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Chapter 7: Problem 37

Solve. $$3+2 x-3=2 x-3$$

Short Answer

Expert verified
No solution; the equation is inconsistent.

Step by step solution

01

Simplify expressions

First, simplify the expressions on both sides of the equation. On the left side, calculate \(3 - 3\) which results in \(0\). Thus, the equation reduces to \(2x = 2x - 3\).
02

Eliminate variable terms on both sides

Subtract \(2x\) from both sides of the equation to remove the \(x\) terms. This simplifies to \(0 = -3\).
03

Analyze the resulting statement

Compare the simplified equation \(0 = -3\). This is a false statement, indicating that there is no solution to the equation.

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Key Concepts

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

Simplifying Expressions
When solving algebra equations, it is important to first simplify the mathematical expressions on both sides of the equation. Simplifying involves performing basic arithmetic operations like addition or subtraction. In our original problem, the left side of the equation has a simple subtraction:
  • First, we had \(3 + 2x - 3\). Simplifying inside the left side parentheses by calculating \(3 - 3\) gives us \(0\), which means our left side becomes \(2x\).
The equation now is reduced from the original \(3 + 2x - 3 = 2x - 3\) to \(2x = 2x - 3\). This simplification helps in identifying what operations to perform next while solving the equation.
Eliminating Variable Terms
The next step in solving equations is eliminating variable terms. Once we have simplified our expressions, our task is to move all variables to one side of the equation. This helps in isolating unknowns if there is a solution. In our case, the equation is \(2x = 2x - 3\). To eliminate the terms with the variable \(x\), we subtract \(2x\) from both sides. This is done by:
  • Subtracting \(2x\) on the left side, transforming it to \(0\).
  • Subtracting \(2x\) on the right side, leaving \(-3\).
After performing this operation, we arrive at the conclusion \(0 = -3\), which points towards the next critical concept鈥攖he existence of a solution for the equation.
No Solution Equations
Sometimes, you may end up with an equation that doesn鈥檛 make sense, like having a statement that is universally false, such as \(0 = -3\). When you get to this point, it means the equation has no solution, also known as a "no solution equation." Here's what that means:
  • In our scenario, the simplified form \(0 = -3\) indicates there is no value of \(x\) that can satisfy the equation, making it impossible to solve for a particular numerical result for \(x\).
  • This happens when the variable terms are canceled out and you're left with a contradiction between constant terms on any side of the equation.
Always remember, if you simplify and remove variables, but are left with a false statement relative to constants, then it's a clear sign that the given equation does not have any valid solutions.

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