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Chapter 5: Problem 4

Solve each equation. $$x-23.3=-4.5$$

Short Answer

Expert verified
The solution to the equation is \( x = 18.8 \).

Step by step solution

01

Understanding the Equation

The equation given is \( x - 23.3 = -4.5 \). Our goal is to solve for the variable \( x \). This means we need to isolate \( x \) on one side of the equation.
02

Identify the Operation to Reverse

In the equation \( x - 23.3 = -4.5 \), we see that \( 23.3 \) is subtracted from \( x \). To isolate \( x \), we need to reverse this subtraction by adding \( 23.3 \) to both sides of the equation.
03

Perform the Addition

Add \( 23.3 \) to both sides of the equation. This gives us: \[x - 23.3 + 23.3 = -4.5 + 23.3\] Simplifying both sides results in: \[x = 18.8\]
04

Verify the Solution

Substitute \( x = 18.8 \) back into the original equation to check the solution:\[18.8 - 23.3 = -4.5\]Simplify the left side: \[-4.5 = -4.5\]Since both sides match, the solution \( x = 18.8 \) is correct.

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Key Concepts

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

Isolation of the Variable
In algebra, isolating the variable is crucial for solving equations like \( x - 23.3 = -4.5 \). The idea here is to have the variable alone on one side of the equation. This process is essential because it allows us to directly identify the value of the variable.There are a few simple rules to follow when isolating a variable:
  • Perform the same operation on both sides of the equation. This keeps the equation balanced.
  • Identify initially what operation is being performed on the variable. This could be addition, subtraction, multiplication, or division.
In our example, the variable \( x \) is being subtracted by \( 23.3 \). To isolate \( x \), we need to reverse this operation.
Inverse Operations
Inverse operations are tools that help in solving equations by canceling out the effects of each side. These operations are the 'undo' functions in math.Here's how they work:
  • If a variable is being added to a number, you subtract that number to undo the addition. Similarly, if a variable is being subtracted, you add the number.
  • If a variable is multiplied by a number, you can divide by that number to reverse the multiplication. If it's divided, you'll multiply.
For our equation, \( x - 23.3 = -4.5 \), since \( 23.3 \) is subtracted from \( x \), we use the inverse operation, which is addition. Adding \( 23.3 \) to both sides neutralizes the subtraction, leading to \( x = 18.8 \).
Checking Solutions
Once we think we've found the solution to an equation, it's always a good idea to check our work. This confirms that the value we've found truly satisfies the original equation.Checking involves:
  • Substituting the calculated value back into the original equation.
  • Ensuring both sides of the equation are equal after substitution.
In this case, substituting \( x = 18.8 \) back into the original equation \( 18.8 - 23.3 = -4.5 \) perfectly balances the equation, confirming our solution is correct. This step ensures there are no errors and confirms our understanding.

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