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Chapter 7: Problem 19

Solve. $$1+12 x=12 x-2$$

Short Answer

Expert verified
There is no solution, as the equation is inconsistent.

Step by step solution

01

Identify Like Terms

Look at both sides of the equation and identify like terms. The equation is \(1 + 12x = 12x - 2\). Both sides have the term \(12x\).
02

Eliminate \(12x\) from Both Sides

We subtract \(12x\) from both sides to eliminate it from the equation. This simplifies the equation to \(1 = -2\).
03

Analyze Result

After simplifying, the equation becomes \(1 = -2\). This is a false statement, indicating there is no solution to this equation.

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Key Concepts

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

Equation Solving
Solving equations is like finding the missing piece of a puzzle. The main goal is to isolate the variable on one side of the equation. When you have an equation, you perform operations to keep it balanced, just like a scale. Think of both sides of the equation as either side of a scale. Whatever you do to one side, you must do to the other to maintain equilibrium.
  • Start by identifying the like terms. Like terms are terms with the same variable raised to the same power. They can be combined or cancelled out to simplify the equation.
  • Perform operations: Addition, subtraction, multiplication, and division can be used to simplify the equation and isolate the variable.
  • Check your work: Once you think you have a solution, plug it back into the original equation to verify it holds true.
Equation solving requires patience and practice, but once you understand the basic rules, it opens a wide range of possibilities in algebra.
False Statement
A false statement in mathematics is a statement that is incorrect or untrue. When solving an equation, if you reach a point where both sides of the equation are not equal, you have encountered a false statement. An example of this is when simplifying the equation led to 1 = -2.
  • Mathematically Impossible: The statement "1 = -2" contradicts fundamental arithmetic principles. It is simply not true and indicates an error elsewhere if you're solving an equation expecting a valid solution.
  • Clues for Errors: If you reach a false statement inadvertently during calculations, it often suggests a mistake along the way or special considerations in the problem.
Recognizing false statements is crucial in determining the nature of algebraic problems, particularly when exploring the possibility of no solutions.
No Solution Cases
In algebra, some equations have no solution. This occurs when any attempt to solve the equation leads to impossible or false statements. These are known as "no solution" cases.
  • Contradiction Example: The result 1 = -2 is a classic example of a contradiction that suggests an equation is unsolvable under conventional means.
  • Understanding Red Flags: Equations where variables cancel out and leave unequal constants, like our example, typically have no solution.
  • Special Scenarios: Sometimes equations appear balanced before solving but have elements that mathematically contradict each other upon simplification.
Identifying and understanding cases of no solution involves recognizing these peculiar patterns and ensures clarity in problem-solving approaches.

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