Problem 29 - Prealgebra

Question

Simplify each side of the following equations before applying the addition property. $$x-6+4=-3-2$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -3\).

Step 1: Simplify the Left Hand Side (LHS)

Examine the left hand side of the equation, which is given as \(x - 6 + 4\). Combine the constant terms: \(-6 + 4 = -2\). Therefore, the simplified LHS is \(x - 2\).

Step 2: Simplify the Right Hand Side (RHS)

Now, look at the right hand side of the equation \(-3 - 2\). Simplify by combining the terms: \(-3 - 2 = -5\). Thus, the simplified RHS is \(-5\).

Step 3: Combine Simplified Sides into an Equation

Combine the simplified expressions from both sides into a new equation. This gives us: \(x - 2 = -5\).

Step 4: Apply the Addition Property of Equality

To isolate \(x\), add 2 to both sides of the equation \(x - 2 = -5\). This yields: \(x - 2 + 2 = -5 + 2\). Simplifying gives \(x = -3\).

Key Concepts

Simplifying Equations

Simplifying equations means making each side of an equation more manageable and easier to understand. In the context of solving equations, simplification is often the first step.

When tasked with simplifying, focus on combining like terms. Like terms are terms that contain the same variable raised to the same power or are just constants.

• In our example, the left side of the equation is expressed as \(x - 6 + 4\).
• To simplify, we combine the constant terms, \(-6\) and \(4\), to get \(-2\).
• This makes the left side of the equation \(x - 2\).

Breaking down the equation in this way helps pave the way for further steps, particularly when performing operations to isolate the variable.

Addition Property

The addition property of equality is a fundamental principle in algebra that allows us to manipulate equations safely. According to this property, you can add the same number to both sides of an equation, and the balance of the equation will not change. This is crucial when solving for a variable.

In our exercise, after simplifying both sides of the equation, we have to remove \(-2\) from the left hand side to isolate \(x\).

• By applying the addition property, we add \(2\) to both sides of the equation: \(x - 2 = -5\).
• This transforms to \(x - 2 + 2 = -5 + 2\).
• As a result, we are left with \(x = -3\).

This method ensures that the equation remains true and allows us to correctly solve for the unknown variable.

Left Hand Side

The left hand side of an equation (often abbreviated as LHS) is the portion of the equation that appears to the left of the equal sign. In algebraic processes, it is essential to handle both sides of the equation with care. In our example, the LHS begins as \(x - 6 + 4\).

• The first task is to simplify this side by bringing together the constants \(-6\) and \(4\) resulting in \(-2\).
• Once simplified, the LHS reads \(x - 2\), making it easier to manage and balance with the other side.

This simplification is critical as it allows for consistency and a free path to solving the equation successfully.

Right Hand Side

Likewise, the right hand side (RHS) is the equation part that appears to the right of the equal sign. The integrity of the solved equation leans heavily on a correct simplification of the RHS.Consider our starting point: \(-3 - 2\).

• Let's simplify by combining these constants, giving us \(-5\).
• This new, simpler expression makes it easier to compare and balance against the left side.

Reaching this simplified form, though seemingly small, is a substantial part of setting up the equation for solving accurately. Every change we make focuses on maintaining a true equation while simplifying the processes.