Problem 36 - Prealgebra

Question

Solve each equation. Check your solution. $$-52.23+b=40.04$$

Step-by-Step Solution

Answer

The solution is \(b = 92.27\). Substituting back verifies the equation.

Step 1: Understand the Problem

The equation we have is \(-52.23 + b = 40.04\). The goal is to solve for \(b\) by isolating it on one side of the equation.

Step 2: Isolate the Variable

To isolate \(b\), we need to add 52.23 to both sides of the equation. This step helps to cancel out \(-52.23\) from the left side.\(-52.23 + b + 52.23 = 40.04 + 52.23\)

Step 3: Simplify the Equation

After adding 52.23 to both sides, the equation simplifies to:\(b = 92.27\)This shows the value of \(b\) that satisfies the equation.

Step 4: Check the Solution

To verify if \(b = 92.27\) is correct, substitute \(b\) back into the original equation:\(-52.23 + 92.27 = 40.04\).Calculate \(92.27 - 52.23 = 40.04\), which is true. Hence, the solution is verified to be correct.

Key Concepts

Isolating the Variable

To solve an equation like \[-52.23 + b = 40.04\], the first key step is to isolate the variable. Here, the variable is \(b\), and our goal is to have it stand alone on one side of the equation. This step is crucial as it allows us to determine the exact value of \(b\).
To start, we need to eliminate the constant from the side of the variable. In this case, we have \(-52.23\) accompanying \(b\). To "move" this number away from \(b\), we do the opposite operation. Since the number is subtracted, we add \(52.23\) to both sides. This operation can be viewed as maintaining balance; whatever you do to one side of the equation, you must do to the other.
After performing this addition, we end up with:

• \(-52.23 + 52.23 + b = 40.04 + 52.23\)

This results in \(b\) being isolated, since \(-52.23 + 52.23\) becomes 0, leaving us with \(b = 92.27\). Now, \(b\) stands alone, giving us the solution for the equation.

Checking the Solution

Once we've solved the equation for \(b\), it's always essential to confirm that our solution is correct. To do this, we substitute the value of \(b\) back into the original equation to see if it holds true.
For our example, we found that \(b = 92.27\). Replacing \(b\) in the original equation, we get:

• \(-52.23 + 92.27 = 40.04\)

Now, perform the calculation:

• \(92.27 - 52.23 = 40.04\)

Since this equation is correct, our solution \(b = 92.27\) is verified. This step helps to ensure there are no mistakes in our calculation or simplification of the equation. It's always a good practice to double-check in math to avoid errors.

Equation Simplification

Simplifying an equation is all about making it straightforward and easy to understand. After isolating the variable, the next step is to simplify both sides of the equation so that it clearly shows the solution.
For our equation:

• \(-52.23 + b = 40.04\)

After adding \(52.23\) to both sides, which cancels \(-52.23\) on one side, the equation becomes:

• \(b = 92.27\)

This simplified equation makes it clear what the value of \(b\) is, without any extra numbers or terms. This not only makes it easier to check the solution but also helps prevent errors in more complex problems.