Problem 28 - Algebra 2

Question

Name the property illustrated by each statement. If \(5+b=13,\) then \(b=8\)

Step-by-Step Solution

Answer

The property illustrated is the Subtraction Property of Equality.

Step 1: Identify the Given Equation

The problem provides us with the equation \(5 + b = 13\). Our task is to identify the mathematical property shown when it is solved to find \(b = 8\).

Step 2: Perform Subtraction to Simplify

To solve for \(b\), we subtract 5 from both sides of the equation: \[ 5 + b - 5 = 13 - 5 \] This simplifies to: \[ b = 8 \] This step uses the Subtraction Property of Equality, which allows us to subtract the same value from both sides of an equation.

Step 3: Identify the Property

The operation of subtracting the same quantity from both sides to maintain equality represents the Subtraction Property of Equality. This property states that if you subtract the same number from both sides of an equation, the equality is preserved.

Key Concepts

Equation Solving

When it comes to mastering the art of equation solving, it's all about balance. An equation is like a perfectly balanced scale. Whatever you do to one side of the equation, you must do to the other to keep it balanced. The main goal in solving equations is to find the value of the variable that makes the equation true.

In our example, the equation is given as \(5 + b = 13\). To solve for \(b\), we need to isolate it on one side. This involves manipulating the equation by performing operations such as addition, subtraction, multiplication, or division to both sides.

Here, subtraction was used to solve for \(b\). Recognizing when to use each operation effectively is key to becoming confident with equation solving.

Algebraic Properties

Algebraic properties govern how we can manipulate equations, ensuring we maintain the truth of the mathematical expressions. These properties are the foundational rules that allow us to solve equations systematically. Among these properties, the Subtraction Property of Equality stands out in our example.

This property states that if you subtract the same number from both sides of an equation, the equality remains true. In short, it's about keeping the equation balanced while simplifying it. By applying this property to the equation \(5 + b = 13\), we subtract 5 from both sides, leaving \(b = 8\).

Other critical algebraic properties include the Addition Property of Equality, the Multiplication Property of Equality, and the Division Property of Equality. Each of these ensures that equations can be simplified or rearranged while maintaining their truth.

Maintaining Equality

Maintaining equality is crucial when working with equations. Every step you take should preserve the balance of the equation, which is essential for finding the correct solution.

In the process of solving equations, maintaining equality means applying the same operation to both sides. If you add, subtract, multiply, or divide by the same number on each side, the equation remains consistent.

In our example, we subtracted 5 from both sides to isolate \(b\), demonstrating the Subtraction Property of Equality. By keeping the equation balanced at all times, we ensure that the steps we take lead us closer to the solution without altering the equation's truth.

• Always perform the same operation on both sides.
• Check your work by plugging the solution back into the original equation.

Remember, maintaining equality is the linchpin of solving equations accurately.