Problem 34 - Elementary Algebra Within Reach

Question

In Exercises 33-38, justify each step of the equation. Then identify any properties of equality used to solve the equation. $$ \begin{aligned} x+4 &=16 \\ x+4-4 &=16-4 \\ x &=12 \end{aligned} $$

Step-by-Step Solution

Answer

The equation \(x+4=16\) is solved using the subtraction property of equality to yield \(x=12\). The properties of equality used here are the subtraction property and reflexive property of equality.

Step 1: Apply Property of Equality

This step involves the subtraction property of equality. The statement \(x+4=16\) means that the quantity \(x+4\) is equal to 16. To make x the subject of the formula, subtract 4 from both sides of the equation to balance it. This gives \(x+4-4=16-4\)

Step 2: Perform the Subtraction

The next step is to perform the subtraction operation on both sides of the equation. This will lead to \(x=12\)

Step 3: Identify the Properties of Equality Used

The properties of equality used in this exercise are the subtraction property, which allows for the same quantity to be subtracted from both sides of an equation, maintaining equality, and the reflexive property, which states that any quantity is equal to itself, as used in the final step \(x=12\).

Key Concepts

Subtraction Property of Equality

The Subtraction Property of Equality is a fundamental concept in algebra. It states that if you subtract the same number from both sides of an equation, the equality remains true. Consider the equation \( x + 4 = 16 \). To isolate \( x \), we subtract 4 from both sides, resulting in \( x + 4 - 4 = 16 - 4 \). This simplifies to \( x = 12 \).
This property is crucial because it ensures that the equation remains balanced. By applying this property consistently, you maintain the logical integrity of the equation. This makes it easier to solve for variables without altering the truth of the equation. Remember:

• Subtract the same number from both sides to remain balanced.
• It is used to isolate variables in equations.

Equation Balancing

Equation balancing is the process of maintaining equality on both sides of an equation while performing operations. When solving equations, it's crucial to treat both sides equally.
This means that any operation (like addition, subtraction, multiplication, or division) must be performed on both sides. For example, starting with \( x + 4 = 16 \), we subtracted 4 from each side resulting in \( x = 12 \), ensuring both sides are treated the same.
Here are some key points:

• Perform identical operations on each side to retain balance.
• Equation balancing helps in simplifying and solving equations accurately.

Reflexive Property

The Reflexive Property is a simple yet profound concept in mathematics which states that any expression or number is always equal to itself, like \( a = a \). In the context of solving equations, this property assures us that once simplified, expressions such as \( x = 12 \) are inherently true due to this property.
When you arrive at a solved equation, the Reflexive Property confirms that the solutions are valid under the assumed operations.
Keep in mind:

• This property reassures the accuracy of equations once solved.
• It's an inherent truth in mathematics, often underlying solution validations.

Elementary Algebra

Elementary algebra involves the basic understanding and manipulation of algebraic expressions and equations. In exercises like \( x + 4 = 16 \), elementary algebra includes understanding how to perform operations, apply properties like subtraction, and ultimately solve for \( x \).
It forms the foundation for more complex algebraic concepts and is vital for problem-solving skills. Knowledge of elementary algebra equips learners with the tools to tackle a wide range of mathematical problems. Key takeaways:

• Focus on understanding fundamental properties and operations.
• Ensure a solid grasp of solving linear equations.

Solving Equations

Solving equations is the process of finding the value of the unknown variable that makes the equation true. In the example \( x + 4 = 16 \), the goal is to find the value of \( x \). By applying properties of equality like subtraction, we simplify the equation to \( x = 12 \).
This process is essential in mathematics, allowing us to determine solutions systematically. Solving equations is all about understanding and applying algebraic principles effectively. Tips for solving equations include:

• Identify steps and properties needed for simplification.
• Ensure operations are applied consistently on both sides.