Problem 3 - Algebra A Combined Function

Question

Solve each equation. See Examples 1 and \(2 .\) $$ 15 x-8=10+9 x $$

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Move variable terms to one side

Subtract \(9x\) from both sides of the equation to get all the variable terms on one side: \[15x - 9x - 8 = 10\] Simplifying the left side gives us: \[6x - 8 = 10\]

Step 2: Move constant terms to the other side

Add \(8\) to both sides to move the constants to the right side of the equation: \[6x - 8 + 8 = 10 + 8\] Simplifying gives us: \[6x = 18\]

Step 3: Solve for the variable

Divide both sides by \(6\) to solve for \(x\): \[x = \frac{18}{6}\] Which simplifies to: \[x = 3\]

Key Concepts

Solving Linear Equations

Linear equations are equations where the variable, usually represented as \(x\), appears with a power of 1. The general form is \(ax + b = c\). These equations can be solved using simple arithmetic operations.To solve a linear equation, the main goal is to isolate the variable on one side of the equation. This process involves rearranging terms, which is often done in multiple steps:

• First, get all terms with the variable on one side of the equation. This usually involves adding or subtracting terms from both sides.
• Second, adjust the constant terms so that they are isolated on the opposite side of the equation from the variable.
• Finally, solve for the variable by performing the necessary arithmetic operation, often division or multiplication, to isolate the variable completely.

Solving linear equations requires careful manipulation, but each step is straightforward. As long as you perform the same operation on both sides, the equation remains balanced and you will move towards finding the solution.

Variable Isolation

Variable isolation is a cornerstone in solving equations, particularly with one variable like in most algebra problems. The objective is to have the variable by itself on one side of the equation to easily determine its value. Here’s how it generally works:Initially, you’ll move all terms involving the variable to one side of the equation. Using the example equation \(15x - 8 = 10 + 9x\):1. Subtract \(9x\) from both sides to consolidate all \(x\) terms on one side.2. This results in an expression like \(6x - 8 = 10\). All variable terms are now on the left side.Next, focus on eliminating any constants from the side with the variable. Add or subtract these until the variable term is isolated. This is demonstrated by adding \(8\) to both sides, yielding \(6x = 18\).Variable isolation simplifies an equation to its core, making it much simpler to identify the solution. It’s a systematic approach that ensures clarity and accuracy.

Algebraic Manipulation

Algebraic manipulation involves applying arithmetic operations like addition, subtraction, multiplication, and division to simplify and solve equations. These operations must be performed carefully and systematically to maintain the equality.Within the example \(15x - 8 = 10 + 9x\), algebraic manipulation is used effectively:

• First, subtract \(9x\) from both sides. This is a typical step to reduce the equation to fewer terms. It ensures you gather all similar terms together.
• Next, after consolidating the variable terms, you add \(8\) to both sides to simplify further and adjust constant terms. This operation moves constant numbers around, necessary for addressing each term properly in the equation.
• Finally, you arrive at the simpler equation \(6x = 18\). Dividing both sides by \(6\) solves the equation. Division is used to decouple the variable from its coefficient, providing a numeric solution for \(x\).

Each algebraic manipulation step brings the equation closer to a manageable, solvable form. Following the arithmetic guidelines helps ensure that the integrity of the equation is intact and the correct solution is achieved.