Problem 105 - Introductory Algebra for College Students

Question

Determine whether the given number is a solution of the equation. $$14-2 x=-4 x+7 ;-2 \frac{1}{2}$$

Step-by-Step Solution

Answer

No, -2.5 is not a solution to the equation \(14 - 2x = -4x + 7\).

Step 1: Understand the Equation

The given equation is \(14 - 2x = -4x + 7\). Our task is to find if -2.5 is a solution for the variable 'x' in this equation.

Step 2: Substitute the Given Value

Substitute -2.5 for 'x' in the equation. This gives us \(14 - 2(-2.5) = -4(-2.5) + 7\).

Step 3: Simplify Both Sides

Performing the multiplication and the subsequent addition or subtraction gives us a simplified equation: \(14 + 5 = 10 + 7\). Further simplifying this results in \(19 = 17\).

Step 4: Check Equality

Analyzing the simplified equation, we see that the two sides are not equal (since 19 is not equal to 17), therefore -2.5 is not a solution to the equation.

Key Concepts

Algebra for College Students

Algebra at the college level often presents a significant step up from the fundamentals taught in high school. The complexities and abstract concepts introduced in college algebra are designed to challenge students and prepare them for advanced mathematical courses. College students must develop a strong foundational understanding in order to successfully navigate through these advanced topics.

One of the key skills to master is the ability to work with equations. Being adept at solving equations not only helps within mathematics but also in various fields that require analytical problem-solving. A solid grasp on solving algebraic equations allows you to dissect problems methodically, understand relationships between variables, and find solutions to real-world issues. In the original exercise, students are tasked with verifying if a given number is a solution to an equation, which is a fundamental practice in understanding how to manipulate and solve algebraic expressions.

Equation Solving Tutorial

The process of solving equations is at the heart of algebra. To ensure a clear understanding of equation solving, it's helpful to approach the problem in an organized manner. One effective strategy is to break the process into clear, actionable steps, as shown in the step-by-step solution provided for the original exercise.

Decoding the Questions

Begin by carefully reading the problem to comprehend what is being asked. Acknowledge the type of equation you are dealing with, and then proceed with the appropriate method.

• Is it linear, quadratic, or something else?
• What value or values are you solving for?
• Is the equation already simplified, or are there simplifications you must perform before proceeding?

Implementing the Solution

Once you’ve understood the question, the next step is the substitution of the given number and simplifying the equation to see if both sides equal the same value. If they do, the number is indeed a solution to the equation.

Substitution Method in Algebra

The substitution method is a powerful tool in algebra, particularly when you're dealing with systems of equations or, as in our exercise, when needing to validate a suspected solution. The essence of the method is to replace a variable with its value and simplifying to draw conclusions.

Applying Substitution

Following the step-by-step solution in the exercise, start by substituting the given number into the equation wherever the variable 'x' appears. Notice that we replace 'x' with -2.5 and then perform the required arithmetic operations, including multiplication and addition or subtraction. This direct and logical approach helps you to avoid errors and facilitates a straightforward path to the evaluation of the equation.

Ultimately, the substitution method is a foundational technique that allows for the systematic verification of potential solutions in an equation. By mastering this technique, you will be better equipped to tackle more complex algebraic challenges down the road.