Problem 31 - Prealgebra

Question

Simplify each side of the following equations before applying the addition property. $$3-5=a-4$$

Step-by-Step Solution

Answer

The solution to the equation is \(a = 2\).

Step 1: Simplify Left Side

On the left side of the equation, simplify the expression \(3 - 5\). This is calculated as follows: \(3 - 5 = -2\). So, the equation becomes \(-2 = a - 4\).

Step 2: Simplify Right Side

The right side of the equation, \(a - 4\), is already in its simplest form. No changes are needed here.

Step 3: Apply Addition Property

To isolate \(a\), use the addition property of equality by adding \(4\) to both sides of the equation. The equation \(-2 + 4 = a - 4 + 4\) simplifies to \(2 = a\).

Key Concepts

Addition Property of Equality

In algebra, the addition property of equality is a fundamental concept that allows us to maintain the balance in an equation while solving it. The essence of this property is quite simple: adding the same number to both sides of an equation keeps the equation true.

Think of an equation like a seesaw, where both sides need to be balanced. If we add weight (or numbers) to one side, we must add the same to the other side to keep it level.

• If you have an equation like \(x - 3 = 7\), and you want to solve for \x\, adding \3\ to both sides results in \(x = 7 + 3\) or \(x = 10\).
• In the exercise, we used this property by adding \4\ to both sides to simplify and find the value of \a\.

This technique helps us isolate variables and solve equations effectively. It's a simple but powerful tool in your math toolkit.

Simplifying Expressions

Simplifying expressions is about making them easier to work with by performing basic arithmetic and combining like terms. It's like cleaning up clutter to see what's really there.

Before we solve equations, it's best to simplify the expressions on each side. This means performing operations like addition, subtraction, multiplication, or division present in the expression.

• In the exercise, we simplified \(3 - 5\) on the left side to get \(-2\).
• Simplification helps us to see the true structure of an expression and prepares it for solving.

By simplifying, you make the math neater and often easier to handle. It’s a crucial first step in solving many algebraic equations.

Prealgebra

Prealgebra is an essential math course that lays the groundwork for understanding more advanced algebraic concepts. It focuses on fundamental operations and basic algebraic skills.

This subject introduces students to concepts such as variables, simple equations, basic arithmetic, and the properties of numbers. Prealgebra serves as a bridge between basic arithmetic and high-level math topics.

• Skills like simplifying expressions and applying the addition property of equality are core components of prealgebra.
• The exercise showcases these skills by starting with numbers and basic operations, leading to solving an equation for a variable.

A strong foundation in prealgebra ensures students are better prepared for algebra and beyond, making it an important phase in one's mathematical education.