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Chapter 3: Problem 30

Use the properties of equality to help solve each equation. $$ 9 x=-108 $$

Short Answer

Expert verified
The solution is \(x = -12\).

Step by step solution

01

Identify the Equation

We are given the equation \(9x = -108\). Our goal is to solve for \(x\).
02

Apply the Division Property of Equality

To isolate \(x\), divide both sides of the equation by 9: \[\frac{9x}{9} = \frac{-108}{9}\] This simplifies to \(x = -12\).
03

Verify the Solution

To ensure the solution is correct, substitute \(x = -12\) back into the original equation: \[9(-12) = -108\] Calculating the left side, we get \(-108 = -108\), which confirms our solution is correct.

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Key Concepts

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

Equation Solving
Solving an equation is a process where we find a value for the variable that makes the equation true. In our example, the equation was given as \(9x = -108\). Here, our goal was to determine the value of \(x\) that would balance both sides of this equation.The process involves manipulating the equation, using various mathematical operations, to isolate the variable. This often requires using properties of equality. In this specific case, once we recognize the form of our equation, our aim becomes to isolate \(x\) on one side of the equation. When completed successfully, you find the solution to the problem, in this instance, \(x = -12\).Key steps include:
  • Identifying the variable that needs to be isolated.
  • Performing operations equally on both sides to maintain the balance.
  • Arriving at a simplified form where the solution is apparent.
Division Property of Equality
The Division Property of Equality is a fundamental principle used in solving equations. It states that if you divide both sides of an equation by the same non-zero number, the two sides remain equal. This property helps us maintain the balance in equations while simplifying them.In our example where \(9x = -108\), we employed this property to solve for \(x\). By dividing both sides by 9, the coefficient of \(x\), the equation reduces to \(x = -12\). This step is crucial because it isolates the variable, making it straightforward to identify the solution.The steps typically involve:
  • Identifying the coefficient of the variable.
  • Dividing both sides of the equation by this number.
  • Ensuring that the division simplifies the equation effectively.
Verification of Solutions
After finding a solution, it鈥檚 important to check your work. Verification ensures that our solution satisfies the original equation. This is a crucial step because it confirms the accuracy and validity of the solution found.In this case, after determining that \(x = -12\) was our solution, we plugged it back into the original equation \(9x = -108\) to test its correctness. Substituting \(-12\) for \(x\), we completed the multiplication \(9(-12)\) and confirmed that the left side equaled the right side, \(-108 = -108\).Why verify? Here are some points:
  • It helps catch calculation errors or incorrect steps.
  • Reassures that the solution is, in fact, a valid answer.
  • Builds confidence in understanding and applying the solving method.

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