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Chapter 3: Problem 31

Use the properties of equality to help solve each equation. $$ -6 x=102 $$

Short Answer

Expert verified
The solution to the equation is \(x = -17\).

Step by step solution

01

Understand the Equation

The equation given is \(-6x = 102\). This equation implies that \(x\) is being multiplied by \(-6\) and the result is \(102\).
02

Identify the Operation Needed

To isolate \(x\), we need to divide both sides of the equation by \(-6\) because \(x\) is currently being multiplied by \(-6\). Dividing by \(-6\) will cancel out the multiplication.
03

Divide Both Sides by -6

Perform the division on both sides of the equation: \[x = \frac{102}{-6}\]
04

Simplify the Expression

Calculate the division \[x = \frac{102}{-6} = -17\].
05

Verify the Solution

Substitute \(-17\) back into the original equation to verify:\(-6(-17) = 102\), which simplifies to \(102 = 102\). The solution is correct.

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Key Concepts

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

Properties of Equality
When solving equations, one of the most fundamental concepts is the **Properties of Equality**. These rules allow us to maintain balance on both sides of an equation. Think of an equation as a balanced scale. Whatever you "do" to one side, you must also "do" to the other to keep it even.

The main properties include:
  • **Addition Property of Equality**: You can add the same number to both sides of an equation, and it will remain equal. For example, if \(a = b\), then \(a + c = b + c\).
  • **Subtraction Property of Equality**: Similarly, subtracting the same number from both sides will also keep the equation balanced. If \(a = b\), then \(a - c = b - c\).
  • **Multiplication Property of Equality**: Here, multiplying both sides by the same non-zero number keeps the equation intact. If \(a = b\), then \(ac = bc\).
  • **Division Property of Equality**: Dividing both sides by the same non-zero number results in an equal equation as well. If \(a = b\), then \(\frac{a}{c} = \frac{b}{c}\), assuming \(c eq 0\).
These properties are crucial because they let us manipulate equations to find unknown values like \(x\) without changing the truth of the original equation.
Division in Algebra
**Division in Algebra** is a key operation, especially when it comes to solving equations. When an unknown variable, such as \(x\), is multiplied by a number, division is often the way we isolate the variable to find its value.

In the given problem, the equation \(-6x = 102\) means that \(x\) is being multiplied by \(-6\). To solve for \(x\), you need to perform the opposite operation, which is division. Here's how:
  • Identify the coefficient you need to divide by, which is \(-6\) in this case.
  • Divide both sides of the equation by \(-6\): \[ x = \frac{102}{-6} \]
  • Perform the division to find \(x\): \[ x = -17 \]
Through division, we isolate \(x\) on one side of the equation, providing a clear solution.

Always remember, division by zero is undefined, so be careful with equations where the divisor might be zero.
Verifying Solutions
Once you solve an equation and obtain a value for the variable, the next important step is **Verifying Solutions**. This ensures that the solution is correct and makes the equation true when substituted back into the original.

In our example, after solving \(-6x = 102\), we found \(x = -17\). To verify:
  • Substitute \(-17\) back into the original equation to replace \(x\).
  • Perform the computation: \(-6(-17)\).
  • Verify that both sides of the original equation equal one another: \(102 = 102\).
This process confirms the accuracy of your solution and reinforces your understanding of algebraic balance.

Always verify your solutions to avoid errors and build confidence in solving equations.

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Most popular questions from this chapter

An executive is earning \(\$ 145,000\) per year. This is \(\$ 15,000\) less than twice her salary 4 years ago. Find her salary 4 years ago.

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Consider these two solutions: $$ \begin{array}{rrr} 3(x+2)=9 & 3(x-4)=7 \\ \frac{3(x+2)}{3}=\frac{9}{3} & 3 x-12=7 \\ x+2=3 & 3 x=19 \\ x=1 & x=\frac{19}{3} \end{array} $$ Are both of these solutions correct? Comment on the effectiveness of the two different approaches.
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