Problem 11 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$4-(z+6)=8$$

Step-by-Step Solution

Answer

The solution to \(4-(z+6)=8\) is \(z = -10\).

Step 1: Simplify the Left Side of the Equation

We'll start by simplifying the left-hand side of the equation by implementing the distribution property to clear the parentheses: \(4-(z+6)\) will simplify to \(4-z-6\), which further simplifies to \(-z-2\). Our equation now is \(-z-2=8\).

Step 2: Solve for z

Our aim is to get \(z\) alone on one side. Thus, we will add \(z\) to both sides, and then subtract 8 from both sides giving us: \(z = -10\).

Step 3: Verifying the Solution

Lastly, to confirm the solution is correct, we will substitute \(z\) with \(-10\) into the initial equation. This yields: \(4-(-10+6)\), which simplifies to \(4-(4)=8-8\), thus 0 equals 0, validifying our solution.

Key Concepts

Distribution Property

When solving equations, the distribution property is a key step. It involves breaking down expressions within parentheses to simplify equations. In this exercise, we had an equation:

• \( 4 - (z + 6) = 8 \)

To simplify it, we applied the distribution property on \( -(z+6) \). By distributing, we change the signs of each term inside the parentheses:

• The \( +z \) becomes \( -z \)
• The \( +6 \) becomes \( -6 \)

This results in the expression:\(-z - 6\). Don't forget the original 4 in front of this expression. Therefore, we have:

• \( 4 - z - 6 \)

Further simplification is done by combining like terms, that is, \( 4 - 6 = -2 \). Thus, the equation simplifies to:

• \( -z - 2 = 8 \)

Applying the distribution property allows us to break down expressions easily and prepare the equation for the next steps.

Verification of Solution

After finding a potential solution for the variable, it’s crucial to check if it's correct. This step ensures our calculations were accurate and the solution satisfies the original equation. In our exercise:

• We solved \( -z - 2 = 8 \) which gave us \( z = -10 \)

To verify this, we substitute \( z = -10 \) back into the original equation \( 4 - (z + 6) \):

• Substitute: \( 4 - (-10 + 6) \)
• Calculate inside the parenthesis: \( -10 + 6 = -4 \)
• So, it becomes \( 4 - (-4) \), which simplifies to \( 4 + 4 \) because subtracting a negative is the same as adding a positive
• This gives \( 8 \), verifying our equation \( 8 = 8 \)

Through these steps, we've confirmed our solution is accurate. Verification is essential as it reassures us that no mistakes were made in the simplification process.

Algebraic Simplification

Algebraic simplification is the process of making equations easier to solve by reducing complexity. We simplify by:

• Combining like terms
• Eliminating unnecessary components

In our equation \( -z - 2 = 8 \), the simplification process involves a few key steps:

• Adding \( z \) to both sides to isolate terms including \( z \)
• That transforms the equation to \( -2 = z + 8 \)
• To find \( z \), subtract 8 from both sides yielding: \( z = -10 \)

These steps move us towards solving the equation by focusing on isolating \( z \). Simplification helps remove unnecessary barriers and guides us toward finding the solution. The main goal here: simplifying to make solving more manageable and intuitive.