Problem 63 - Algebra 2

Question

Name the property illustrated by each statement. If \(3 x=10,\) then \(3 x+7=10+7\)

Step-by-Step Solution

Answer

Addition Property of Equality.

Step 1: Identify the Operations Involved

Examine the statement. It starts with the expression \(3x = 10\) and then transforms it into \(3x + 7 = 10 + 7\). This involves adding the number 7 to both sides of the equation.

Step 2: Recall the Property of Equality

The property that allows us to add the same number to both sides of an equation without changing the equality is the Addition Property of Equality. This property states that if \(a = b\), then \(a + c = b + c\) for any number \(c\).

Step 3: Determine the Property

Since the same number (7) is added to both sides of the original equation without altering its balance, the property being utilized here is the Addition Property of Equality.

Key Concepts

Properties of Equality

The Addition Property of Equality is a special rule in mathematics that helps keep an equation balanced. When solving equations, it is crucial that both sides of the equation remain equal. Imagine both sides of an equation as two pans of a balance scale. By adding the same amount to each pan, the scale remains perfectly balanced.
For example, in the original exercise, we had the equation \(3x = 10\). By adding 7 to both sides, resulting in \(3x + 7 = 10 + 7\), the equality is not affected. This showcases the Addition Property of Equality, reminding us that whatever is done to one side of an equation must also be done to the other.

• Helps maintain balance in equations
• Essential to solving linear equations
• Makes mathematical operations consistent and predictable

Understanding these properties ensures the integrity of our solutions and aids in more effective problem-solving strategies.

Algebraic Operations

Algebraic operations involve the basic mathematical operations of addition, subtraction, multiplication, and division, as applicable to algebraic expressions and equations. These operations are the building blocks for both simplifying expressions and solving equations.
In the context of our exercise, the operation chosen was addition, where we added 7 to both sides of \(3x = 10\). Adding numbers or expressions to both sides of an equation is a fundamental algebraic operation governed by the properties of equality.
Let’s quickly run through other algebraic operations used frequently with equations:

• Addition and Subtraction: Used to isolate variables and solve equations by maintaining balance.
• Multiplication and Division: Applied to simplify equations further by eliminating coefficients attached to variables.

Mastery of these operations, in tandem with understanding the properties of equality, can greatly enhance problem-solving skills.

Solving Equations

Solving equations is a methodical process of finding the value of variables that satisfy a given equation. When equations are balanced, solving them becomes straightforward and reliable. The main objective is to isolate the variable on one side of the equation.
In our example problem, we started with \(3x = 10\) and adjusted it to \(3x + 7 = 17\) using the Addition Property of Equality. If we continued solving for \(x\), we would reverse this operation—subtract 7 from both sides to revert to \(3x = 10\), then divide by 3 to find \(x\).
Here are some basic steps to follow when solving equations:

• Identify the operation needed to isolate the variable.
• Apply the inverse operation to both sides of the equation.
• Simplify each step while maintaining balance.
• Verify the solution by substituting back into the original equation.

Solving equations is a skill refined by practice, and by understanding the fundamental steps and properties involved, you can tackle more complex mathematical challenges with confidence.