Problem 30 - Prealgebra

Question

Simplify each side of the following equations before applying the addition property. $$x-8+2=-7-1$$

Step-by-Step Solution

Answer

The solution is \(x = -2\).

Step 1: Simplify the Left Side

Start by simplifying the expression on the left side of the equation \(x - 8 + 2\). This can be simplified by combining the constant terms \(-8\) and \(+2\). \(x - 8 + 2 = x - 6\)

Step 2: Simplify the Right Side

Simplify the equation on the right side, \(-7 - 1\). These are both constants, so they can be combined directly.\(-7 - 1 = -8\)

Step 3: Set the Simplified Equation

Now, rewrite the equation using the simplified expressions for each side:\(x - 6 = -8\)

Step 4: Apply the Addition Property of Equality

To solve for \(x\), we need to isolate it on one side. Add 6 to both sides of the equation to cancel out the \(-6\) term:\(x - 6 + 6 = -8 + 6\)This simplifies to:\(x = -2\)

Key Concepts

Addition Property of Equality

The Addition Property of Equality is a fundamental concept in algebra. It states that if you add the same number to both sides of an equation, the equality remains true. This is a critical tool for solving equations.

Here's how it works:

• If you have an equation like \( x = 5 \), adding 3 to both sides gives you \( x + 3 = 5 + 3 \), which simplifies to \( x + 3 = 8 \). The equality is still maintained.
• The idea is that as long as you perform the same operation on both sides, the equation stays balanced.

When solving the equation \( x - 6 = -8 \), the goal is to isolate \( x \). To do this, we applied the addition property by adding 6 to both sides of the equation, which cancels out the \(-6\) on the left, leaving us with \( x = -2 \). It's an efficient method to solve linear equations.

Simplifying Expressions

Simplifying expressions in algebra means making them easier to work with. This often involves combining like terms or reducing complex terms to simpler ones. In the equation we are working with, simplifying is the first step in solving it.

• On the left side, we combined \(-8 + 2\) to get \(-6\), resulting in the expression \(x - 6\). This combines all like terms, which makes the expression simpler and more manageable.
• On the right side, combining \(-7 - 1\) simplifies directly to \(-8\).

After simplifying, both sides of the equation become straightforward, allowing us to apply further steps like the Addition Property of Equality clearly. Simplification is crucial because it reduces complexity and sets the stage for easier problem solving.

Prealgebra

Prealgebra is the foundation for all advanced mathematics. It covers basic concepts such as operations with numbers, working with variables, and understanding simple equations. Learning prealgebra equips students with the skills needed to tackle algebra successfully.

In our exercise, prealgebra skills are crucial:

• Understanding how to simplify expressions by combining like terms draws heavily on prealgebra concepts.
• The addition property, used to isolate the variable \(x\), also relies on knowing basic arithmetic and the properties of equality studied in prealgebra.

Through mastering prealgebra, students can develop a strong mathematical foundation, making it easier to understand and solve more complex equations in algebra and beyond. It is about building confidence and competence in handling mathematical expressions and equations.