Problem 20 - Algebra for College Students

Question

Solve each equation. \(2 x-1=6 x+15\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = -4\).

Step 1: Move Variables to One Side

Subtract \(6x\) from both sides to start isolating the variable \(x\). The equation becomes:\[2x - 6x - 1 = 6x - 6x + 15\] This simplifies to \(-4x - 1 = 15\).

Step 2: Isolate the Variable Term

Add \(1\) to both sides to isolate the term with \(x\). The equation becomes: \[-4x - 1 + 1 = 15 + 1\] Simplifying gives \(-4x = 16\).

Step 3: Solve for the Variable

Divide both sides of the equation by \(-4\) to solve for \(x\). The equation is now: \[\frac{-4x}{-4} = \frac{16}{-4}\] Simplifying yields \(x = -4\).

Key Concepts

Solving Linear Equations Step-by-Step

Solving linear equations is like finding a path to help you figure out what an unknown number, called the variable, should be. These equations often look like a balance scale, where what you do to one side, you must do to the other. Let's break down the process.

• Identify the equation: Start by recognizing the structure. Here, we have something like 2x - 1 = 6x + 15. Each part contains terms and numbers.
• Assign values and terms accordingly: Numbers can stand alone or be part of a term like 2x, representing a value you need to find.

Start by moving variables to one side. Subtract or add terms to both sides to keep the equation balanced. It's like moving items on a scale but ensuring each side remains even. As you go through this process, the equation will slowly transform, guiding you toward the solution.

The Power of Isolation of Variables

In solving equations, focusing on getting the variable alone, or isolating it, is crucial. The isolated variable is our target to find its value easily.

• Start by reorganizing the equation: Turn 2x - 1 = 6x + 15 into -4x - 1 = 15. Here, we have moved all terms with x to one side.
• Make the equation simple: Add or subtract constants (like numbers without variables) to isolate the 'x'-term. It changes step-by-step and becomes easier each time.

With the main variable isolated, it becomes much clearer to see what the value of x should be. Think of isolation as clearing out distractions to focus on what truly matters: finding x.

Algebraic Manipulation Explained

Algebraic manipulation is like re-arranging a puzzle until everything fits perfectly. It's the key to solving equations without any guesswork. Here’s how it works:

• Use addition or subtraction: Shift terms across the equation to consolidate similar items. For example, moving the 2x term by subtracting it from 6x.
• Divide or multiply for results: When you reach a stage like -4x = 16, divide both sides by -4. This divides the coefficient, simplifying the equation.
• Check your solution: After manipulation, always ensure your answer makes sense by substituting back into the original equation if needed.

Algebraic manipulation allows you to deftly navigate through complexities. It's a skill that applies the magic of balance and arithmetic to untangle even the knottiest mathematical problems.