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Chapter 3: Problem 37

Solve the equation. $$12-6=-n$$

Short Answer

Expert verified
The solution of the equation is \(n=-6\).

Step by step solution

01

Simplify the left-hand side

Start by simplifying the left-hand side of the equation. The expression \(12-6\) equals 6. Therefore, the equation becomes \(6=-n\).
02

Change the sign of the variable's coefficient

Currently, 'n' has a negative sign in front. To solve for 'n', we need to change the sign of the variable's coefficient. We do this by multiplying the right-hand side by -1. The equation becomes \(-6=n\).

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Key Concepts

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

Simplification
Simplification is an essential step when solving equations. It helps in making the equation easier to understand and work with. In our example, the equation starts as \(12-6=-n\). The task here is to simplify the left-hand side of the equation.
  • First, identify what operations need to be performed. Here, we subtract 6 from 12.
  • Calculate the result of the subtraction: \(12-6=6\).
  • Replace the original expression with the simplified result, leading to the equation \(6 = -n\).
Simplifying before solving makes the equation cleaner and reduces the likelihood of mistakes. This step highlights the essence of dealing with numbers separately before addressing the variables.
Equation Solving Steps
Solving an equation involves clear, logical steps aimed at isolating the variable to find its value. Consider these steps as a roadmap to reach the "solution." For the equation \(6=-n\), follow these steps:
  • Write down the equation clearly without changing its form unintentionally.
  • As discussed, simplify any expressions present to make the equation less complex.
  • Focus on getting the variable 'n' by itself on one side of the equation. This means performing operations that will leave 'n' alone.
  • By multiplying by -1, transform \(-n\) into \(n\) in this context.
  • Check your work by substituting back into the initial scenario, verifying that your solution works correctly.
These steps ensure that each part of the equation is addressed methodically, leading to successful solving of the problem. Breaking down the equation into manageable parts makes the entire process smoother.
Changing Variable Coefficient Sign
Sometimes, when solving equations, a variable comes with a negative sign attached to its coefficient. This can make working with the variable challenging because our goal is to find its positive or actual value. To deal with a negative coefficient, just like in our example with \(6 = -n\), we follow a simple method:
  • Identify the side where the variable with the negative sign is present. Here, it's on the right side.
  • Change the sign of the coefficient by multiplying by -1. This transforms \(-n\) into \(n\) and \(6\) into \(-6\).
  • Rewrite the simplified equation: it now reads \(-6 = n\).
Remember, flipping the sign when needed helps reveal the correct value of 'n'. This small step is crucial for ensuring you reach the right solution without extra complications.

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