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Chapter 6: Problem 40

Solve and check. $$x-5.8=4.11$$

Short Answer

Expert verified
Question: Solve the linear equation: x - 5.8 = 4.11 Answer: x = 9.91

Step by step solution

01

Add 5.8 to both sides of the equation

Begin by adding 5.8 to both sides of the equation to isolate the term containing x. $$x - 5.8 + 5.8 = 4.11 + 5.8$$
02

Simplify the equation

After performing the addition, simplify the equation: $$x = 9.91$$
03

Check the solution

Now, substitute our value of x (9.91) back into the original equation to verify that it's correct. $$9.91 - 5.8 = 4.11$$ Which simplifies to: $$4.11 = 4.11$$ Since both sides of the equation are equal, this confirms that our solution is correct. The value of x is 9.91.

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Key Concepts

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

Understanding Algebraic Equations
Algebraic equations are like puzzles where you find the value of the unknown, usually represented by a letter, like \(x\). In the equation \(x - 5.8 = 4.11\), we are solving for \(x\) by finding the number that makes the equation true. Key parts of an algebraic equation:
  • Unknown variable: The letter, often \(x\), stands for a number we need to find.
  • Constants: These are the numbers that stay the same, like \(-5.8\) and \(4.11\).
  • Operations: These are the mathematical procedures, such as addition or subtraction, that connect the variable and constants.
Solving equations involves performing operations to isolate \(x\) on one side, which we'll discuss further in the following sections.
Checking Solutions for Algebraic Equations
After solving for \(x\), checking your solution is crucial. It ensures that the value found truly satisfies the original equation. Here's how you can check a solution:
  • Substitution: Replace the variable in the original equation with the solution found. For instance, substitute \(x = 9.91\) back into the equation \(x - 5.8 = 4.11\).
  • Simplification: Perform the operations indicated in the equation with the provided value to verify if both sides equal. In our case, \(9.91 - 5.8\) simplifies to \(4.11\), confirming our solution is accurate since both sides match.
Checking confirms that the answer makes the original statement true, preventing errors and solidifying understanding in solving equations.
Simplification of Expressions
Simplification is the process of making an algebraic expression easier to work with. This is essential when solving equations because it helps clearly isolate the variable. To simplify, follow these steps:
  • Combine like terms: Add or subtract numbers directly related to the variable. In our problem, \(-5.8 + 5.8\) simplifies to \(0\), leaving \(x\) by itself.
  • Perform arithmetic operations: After isolating \(x\), calculate the remaining arithmetic expression on the other side. Thus, \(4.11 + 5.8\) results in \(9.91\).
Simplified expressions make it more straightforward to see the steps taken to solve the equation. In doing so, it demonstrates clearly how calculations determine the value of \(x\).

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