Problem 18 - Algebra 1

Question

Solve the equation. $$3 x-1=8$$

Step-by-Step Solution

Answer

The solution to the equation \(3x - 1 = 8\) is \(x = 3\).

Step 1: Isolate the term with x

The first step in solving the equation \(3x - 1 = 8\) is to isolate the term with x by getting rid of the number (-1) next to it. This can be done by adding 1 on both sides of the equation. This will result in the equation: \(3x = 8 + 1\).

Step 2: Perform the addition

Performing the addition on the right side of the equation gives us: \(3x = 9\).

Step 3: Divide by the coefficient of x

Finally, we divide both sides of the equation by the coefficient of x which is 3 to get the value of x: \(x = 9 / 3\).

Key Concepts

Solving Equations

Solving equations is like solving a puzzle. You want to find out what value of the variable makes the equation true. In the provided exercise, we're given the equation, \( 3x - 1 = 8 \). To solve it, the goal is to get "\( x \)" by itself on one side of the equation. This process is known as isolating the variable.

To begin, think of the equation as balanced scales—whatever you do to one side, you must also do to the other to keep it balanced. Your job is to perform operations that will simplify the equation step-by-step, focusing on isolating the variable "\( x \)" to read exactly with '\( x = ... \)'.
Breaking this down into manageable steps is the key. In our journey of solving, you'll use basic arithmetic: addition, subtraction, multiplication, and division, to untangle the equation systematically.

Algebraic Manipulation

Algebraic manipulation involves using different operations to simplify or rearrange equations. Let's break it down:

• **Addition and Subtraction:** Remove or move terms across the equation to start isolating the variable. For example, to eliminate "-1" from "\(3x - 1 = 8\)", you add 1 to both sides, turning it to "\(3x = 9\)".
• **Multiplication and Division:** Once the term with the variable "\(3x\)" is isolated, you must eliminate the coefficient (here 3) by performing the opposite operation. Divide both sides by 3 to simplify the equation to "\(x = 3\)".
  

Remember, whatever operation you choose, do it equally on both sides to maintain the equality. These steps are fundamental in algebra and crucial for solving more complex equations as you advance.

Step-by-step Solutions

A step-by-step solution is meant to help you understand each phase of the process. It shows you how to solve the equation methodically:
*Step 1: Isolate the Term with X*
Begin by addressing terms not attached to the variable. Add or subtract these numbers to move them to the opposite side of the equation. For \(3x - 1 = 8\), adding 1 to both sides changes it to \(3x = 9\).
*Step 2: Solve for X*
Once the variable term is isolated, divide by the variable's coefficient to solve for it, turning "\(3x = 9\)" into "\(x = 3\)".
These steps are clear and systematic, showing the most direct path to find the solution. They're invaluable in understanding not just this problem, but any linear equation you might encounter.