Problem 51 - Algebra 1

Question

Solve the equation. $$ 2(z-3)=12 $$

Step-by-Step Solution

Answer

The solution to the equation is \(z = 9\)

Step 1: Simplify the equation on left side

Multiply through the parentheses on the left side of the equation: \(2 \times z - 2 \times 3\), which simplifies to \(2z - 6\). Our equation now looks like this: \(2z - 6 = 12\)

Step 2: Transpose

Next, add 6 to both sides of the equation to isolate 2z on one side. This results in the equation \(2z = 18\)

Step 3: Isolate the variable

In order to solve for z, divide each side of the equation by 2. This results in \(z = 9\)

Key Concepts

Solving Equations

Solving equations is all about finding the value of the variable that makes the equation true. Let's break it down. An equation is like a balance scale. What you do to one side, you must do to the other. For example, in our given problem, we start with the equation \(2(z-3)=12\).

• First, distribute the 2 across the parentheses on the left-hand side, multiplying both \(z\) and \(-3\) by 2.
• This gives us \(2z - 6\), simplifying the expression on that side.
• The equation changes to \(2z - 6 = 12\), essentially balancing both sides.
• Next, to further simplify and solve for \(z\), we add 6 to both sides, a critical step maintaining the balance, resulting in \(2z = 18\).

Equation solving often involves these steps of simplification, transposition, and isolation. This provides a structured method to find out the unknown variable you've been tasked to find. Remember, each manipulation keeps the equation balanced.

Variables in Equations

Variables are symbols used to represent unknown numbers in equations. Think of a variable as a placeholder for a number we are trying to find. In the example equation, our variable is \(z\). Here's how we think about variables:

• They allow us to set up equations to model real-world situations.
• A variable can change or vary depending on the conditions or equations.
• When solving, we're essentially peeling away layers to figure out what single number the variable stands for.

In the equation \(2(z-3) = 12\), \(z\) represents this unknown. Solving for \(z\) means simplifying and rearranging the equation until we express \(z\) explicitly by itself.

Simple Arithmetic Operations

Simple arithmetic operations form the backbone of solving any mathematical problem. In our equation, we use addition, subtraction, and multiplication. Here are the steps broken down with arithmetic insights:

• Start with multiplication when simplifying \(2(z-3)\) by distributing: \(2 \times z\) and \(- 2 \times 3\).
• The equation simplifies to \(2z - 6 = 12\). The subtraction \(-6\) is countered by adding 6 to both sides, moving us to \(2z = 18\).
• Lastly, division comes into play. Divide both sides by 2 to isolate \(z\), resulting in \(z = 9\).

Each arithmetic operation builds upon the previous step, methodically leading to the solution. Understanding these operations helps in efficiently tackling more complex equations down the line. So, mastering these basics is vital for mathematical fluency.