Problem 72 - Algebra 2

Question

Solve each equation. $$ 7 z-4=5 z+8 $$

Step-by-Step Solution

Answer

The solution is \(z = 6\).

Step 1: Move Variables to One Side

First, we want to isolate the variable terms on one side of the equation. Subtract \(5z\) from both sides: \[7z - 5z - 4 = 5z - 5z + 8\]This simplifies to: \[2z - 4 = 8\]

Step 2: Isolate the Variable Term

Now, isolate the term containing \(z\) by adding 4 to both sides of the equation:\[2z - 4 + 4 = 8 + 4\]This simplifies to:\[2z = 12\]

Step 3: Solve for the Variable

Finally, divide both sides by 2 to solve for \(z\): \[\frac{2z}{2} = \frac{12}{2}\]This gives:\[z = 6\]

Key Concepts

Solving Linear Equations

Linear equations are algebraic expressions that equate two expressions that are first-degree polynomials. The goal is to find the value of the unknown variable that makes the equation true. In our problem, given equation is:\[7z - 4 = 5z + 8\]To solve linear equations like this, follow structured rule-based techniques to isolate the variable. By applying consistent and logical steps, you can systematically find the solution. The break-down into each manageable part ensures clarity and accuracy. Understanding the underlying rules helps you approach similar problems in a variety of settings.

Equation Isolation Techniques

An essential concept in algebra is isolating the variable term. The initial equation contains variable terms on both sides:- Start by moving all variables to one side of the equation. In this case, subtract \(5z\) from both sides, which makes the variable terms gather: \[7z - 5z - 4 = 8\] Simplifying gives you: \[2z - 4 = 8\]- Next, focus on the term that does not contain the variable. Add or subtract these numeric values to work towards isolation. Here, add 4 to both sides: \[2z - 4 + 4 = 8 + 4\] Which simplifies to: \[2z = 12\]Moving terms across the equality sign fundamentally changes their sign. Always balance this by performing the same operation on both sides to maintain equality.

Variable Manipulation

Variable manipulation is the process of simplifying and rearranging equations to isolate and solve for the unknown variable. Once you've gathered all variable terms on one side, determine the coefficient of \(z\), which here is 2. Divide both sides of the equation by this coefficient:\[\frac{2z}{2} = \frac{12}{2}\]- Simplify to find: \[z = 6\]Remember:- Operations such as addition, subtraction, multiplication, and division alter equations' structure. It's essential to do them evenly on both sides to keep the equation's balance.- Ensure each step leads towards simplifying the equation and making the expression for the variable clear.This systematic breakdown into achievable steps turns complex problems into manageable parts, ensuring success in finding the right solution.