Problem 6 - Introductory Algebra for College Students

Question

Solve each equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. $$3 x+2-x=6+3 x-8$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 6\).

Step 1 Simplify the equation

To simplify the equation, group the like terms together. Starting with the terms of \(x\): \(3x - x = 3x - 6 \). This simplifies to: \(2x = 3x - 6 \).

Step 2 Rearrange the equation

The aim is to make \(x\) the subject of the equation. To do this, subtract \(3x\) from both sides of the equation: \(2x - 3x = 3x - 3x - 6 \). This simplifies to: \(-x = -6 \).

Step 3 Solve for \(x\)

To get the value of \(x\), multiply both sides of the equation by -1: \(-1*-x = -6*-1 \). This simplifies to \(x = 6 \).

Step 4 Verify your solution

To confirm that \(x = 6\) is indeed a solution, substitute \(x = 6\) back into the original equation: \(3*6 + 2 - 6 = 6 + 3*6 - 8 \). The left side simplifies to 14 while the right side simplifies to 14 as well. Therefore, \(x = 6\) is the correct solution as both sides of the equation are equal.

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is crucial when it comes to solving linear equations. An algebraic expression is a mathematical phrase that can include numbers, variables (like x or y), and operation symbols like addition (+), subtraction (-), multiplication (*), and division (÷). When solving equations, identifying and combining like terms is vital. Like terms are terms that contain the same variable raised to the same power. For example, 3x and x are like terms because they both contain the variable x to the power of one.

In the given exercise, 3x and -x on the left and 3x on the right side are like terms. Grouping them simplifies the equation, making the subsequent steps more manageable. It's essential to recognize and correctly manipulate these expressions to progress toward the solution.

Equation Simplification

Equation simplification is a systematic process aimed at making an equation easier to solve. This involves combining like terms and moving terms to different sides of the equation to isolate the variable. When you're presented with an equation like 3x + 2 - x = 6 + 3x - 8, you start by simplifying both sides independently.

Remember to follow the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This is often abbreviated as PEMDAS. Once like terms are combined and the equation is simplified, you move to arranging the equation so that the variable you are solving for is on one side of the equation. In the problem, we subtract 3x from both sides to isolate x, which is an essential step in the puzzle.

Verification of Solutions

Verification is the final, yet a fundamental step in the process of solving linear equations. It ensures the solution obtained is accurate. After finding the proposed value for the variable, in this case x = 6, it's crucial to substitute this value back into the original equation to verify if it satisfies the equation.

For our problem, substituting x = 6 into 3x + 2 - x should yield the same result as substituting it into 6 + 3x - 8. This confirmation step is a good practice not only to check your work but to deepen the understanding of the concept that the solutions to equations are values that maintain equality between the two sides. Always take the time to verify the solution to avoid any errors and ensure full comprehension of the topics covered.