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Chapter 3: Problem 22

Use the properties of equality to help solve each equation. $$ b+0.27=0.74 $$

Short Answer

Expert verified
The solution is \( b = 0.47 \).

Step by step solution

01

Identify the Equation

The given equation is \( b + 0.27 = 0.74 \). We need to solve for \( b \) by isolating it on one side of the equation.
02

Apply the Subtraction Property of Equality

Subtract 0.27 from both sides of the equation to isolate \( b \). This gives us the equation \( b + 0.27 - 0.27 = 0.74 - 0.27 \).
03

Simplify Both Sides

Simplify the left side: \( b + 0 = b \). Simplify the right side by calculating \( 0.74 - 0.27 = 0.47 \).
04

Solution

Now that both sides are simplified, we have \( b = 0.47 \). This is the value of \( b \) that satisfies the equation.

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Key Concepts

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

Solving Equations
When you're solving equations, the goal is to find the value of the unknown variable that makes the equation true. Equations are like balancing acts, similar to weighing scales, where both sides need to stay equal. If you perform an operation on one side, such as adding or subtracting, you must do the same to the other side. This maintains balance, keeping the equation true.
For our given problem, we were asked to solve the equation \( b + 0.27 = 0.74 \). Here, you're determining the number that, when added to 0.27, equals 0.74. By following structured steps, you can confidently solve for the unknown and check your solutions by substituting them back into the original equation.
Subtraction Property of Equality
The Subtraction Property of Equality is a pivotal concept when solving equations. If you subtract the same number from both sides of an equation, the equality remains true.
In the equation \( b + 0.27 = 0.74 \), you want to isolate \( b \). To do this, subtract 0.27 from each side, which leaves us with \( b = 0.74 - 0.27 \).
  • Performing subtraction on both sides ensures each side remains balanced and equal.
  • This allows you to simplify the equation further, bringing you closer to finding the value of the unknown variable \( b \).
  • Being systematic with subtraction helps prevent errors and ensures clarity at each step.
Employing this property correctly is fundamental to working through equations smoothly and effectively.
Isolating Variables
Isolating the variable is often the main goal in solving equations. This means rearranging the equation such that the variable you're solving for is on one side, ideally by itself.
In our example with the equation \( b + 0.27 = 0.74 \), the aim is to find \( b \) by itself on one side of the equation. Use operations like subtraction to simplify the equation: subtract 0.27 from both sides to get \( b = 0.47 \).
  • Isolating the variable helps you directly determine its value.
  • This process usually involves reversing any operations applied to the variable, such as addition or subtraction.
  • Once the variables are isolated, solving becomes a straightforward task.
Through this method, you not only solve the equation but also gain a deeper understanding of the relationships between the numbers involved.

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