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Chapter 2: Problem 55

Solve. $$ -38 x=15,542 $$

Short Answer

Expert verified
The solution for the equation is \(x = -409\).

Step by step solution

01

Identify the Equation

The given equation is \[-38x = 15,542.\] Our goal is to isolate \(x\) so that we can find its value.
02

Rearrange for x

To isolate \(x\), we need to divide both sides of the equation by -38. This gives us \[x = \frac{15,542}{-38}.\]
03

Calculate the Division

Now calculate the division, \[x = \frac{15,542}{-38} = -409.\] Use a calculator to ensure accuracy.

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Key Concepts

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

Step-by-step solution
When dealing with linear equations, using a step-by-step solution approach helps in organizing the process, making it easier to solve the problem accurately. Follow each phase carefully.
  • Start by correctly identifying and rewriting the equation. In this case, it is \-38x = 15,542\.
  • Break down the steps needed to find the solution. These may include isolating the variable, performing calculations or checking your result.
  • Each step should bring you closer to solving for the desired variable, in this case, \(x\).
By breaking down the problem into logical steps, you avoid mistakes and ensure you understand each phase necessary for solving the equation.
Steps should be clear and straightforward, applying consistent logic to arrive at the final answer.
Equation rearrangement
Equation rearrangement is a critical skill in algebra. It involves manipulating an equation to isolate the variable you are trying to solve for. Here's how we can rearrange the given equation:
  • Start with the original equation: \ -38x = 15,542 \.
  • To isolate \(x\), you need to perform the opposite operation of what's currently being done to it. In this equation, \(x\) is being multiplied by \(-38\).
  • To counter this multiplication, you should divide both sides of the equation by \(-38\). This balances the equation, keeping it equal on both sides.
This technique not only simplifies the problem but also helps in understanding the relationship between the different components of the equation.
It transforms the process into manageable chunks, making it easier for the solver to focus on each part.
Division in equations
Division plays a crucial role in solving equations by helping to isolate variables. Understanding division in equations can be particularly helpful when variables are multiplied by a number.
  • In our equation, once we divide by \(-38\), the negative sign is also critical as it affects the result. Thus, \(x = \frac{15,542}{-38}\).
  • Performing division accurately ensures that the variable \(x\) is isolated correctly, giving you the solution: \(-409\).
  • Using a calculator can help avoid mistakes, especially with larger numbers or when the division is not straightforward.
Division in equations not only helps in rearranging the terms but also makes the solution more straightforward after isolating the variable.
By practicing this concept, students can enhance their problem-solving skills and build confidence in dealing with algebraic expressions.

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