Problem 95 - Introductory and Intermediate Algebra for College Students 4th

Question

Writing about mathematics will help you to learn mathematics. For all writing exercises in this book, use complete sentences to respond to the questions. Some writing exercises can be answered in a sentence; others require a paragraph or two. You can decide how much you need to write as long as your writing clearly and directly answers the question in the exercise. Standard references such as a dictionary and a thesaurus should be helpful. How do you determine whether a given number is a solution of an equation?

Step-by-Step Solution

Answer

To determine whether a given number is a solution to an equation, you substitute the number in place of the variable in the equation. If the resulting equation holds true (left-hand side equals right-hand side), then the number is a solution. If not, it isn't a solution.

Step 1: Understanding What A Solution Is

A solution of an equation is a value that satisfies the equation when it is substituted for the unknown. In other words, when you substitute the proposed solution into the equation, the equation comes out as a true mathematical statement.

Step 2: The Substitution Process

To verify whether a given number is a solution of an equation, take that number and substitute it into the equation in place of the variable. It involves replacing the variable in the equation with the number. You can use pencil and paper for the substitutions for better accuracy.

Step 3: Check if the Equation Holds True

After substituting the value into the equation, simplify both sides of the equation if needed and check whether both sides are equal. If they are equal, then the given number is a solution to the equation. If they are not equal, then the given number is not a solution.

Key Concepts

Verification of Solutions

When dabbling in the realm of algebra, it's crucial to understand how to determine if a given number truly solves an equation. Imagine you're a detective examining clues to solve a mystery. Similarly, in math, verification is like finding conclusive evidence that supports your hypothesis. The process begins simply by taking the suspected solution and plugging it into the original equation, replacing the variable. This hands-on moment is revealing; if both sides of the equation balance perfectly after the substitution, then you've confirmed the solution. If not, the search continues. It's essential to carry out each step with care, ensuring every operation adheres to mathematical rules. This confirmation process bolsters your understanding and trust in the equations you work with.

For example, to check if 5 is a solution of the equation \(2x + 3 = 13\), you'd substitute 'x' with 5, resulting in \(2(5) + 3\). The left side becomes 13, which matches the right side, confirming that x=5 is indeed a solution. But imagine if you get different numbers on each side of the equation after substitution, like in checking if 4 is a solution to the same equation and getting \(2(4) + 3 = 11\), which clearly doesn't match the 13. You've just proven that 4 is not a solution. This method keeps the guesswork out of algebra and grounds your work in logic and evidence.

Substitution Method

Substitution stands as one of the foundational techniques in solving algebraic equations. Think of it as the secret ingredient in a recipe that brings the whole dish together. In mathematics, the substitution method involves replacing variables with known or given values to simplify problems and find solutions. It is like swapping out pieces in a puzzle until you see the complete picture.

Picture you are working with the equation \(y = 2x + 4\) and another equation \(x + y = 10\). By using the substitution method, you can take the expression from the first equation and put it in place of 'y' in the second equation, yielding \(x + (2x + 4) = 10\). Rather than juggling two unknowns, you now have one equation with one unknown, which feels much more manageable. Simplify it to \(3x + 4 = 10\), and you're well on your way to finding the value of 'x'. Always remember to check your work by plugging the solution back into the original equations to ensure it fits snuggly into place.

Mathematical Problem-Solving

Navigating through mathematical problem-solving is akin to embarking on a strategic expedition, where each step is deliberately chosen to ensure success. The art of solving mathematics problems spans beyond mere computation; it involves a logical sequence of steps underpinned by critical thinking and a deep understanding of concepts.

Initially, define the problem: What exactly is being asked? Then, devise a plan: Which strategies or techniques can you apply? For instance, equations might call for different tactics – like graphing, factoring, or using the aforementioned substitution. Execute the plan with precision, showing each step to make it reproducible and checkable. Finally, critically review your solution. This cycle enhances your problem-solving skills immeasurably and prepares you for a variety of future mathematical challenges. To illustrate, if tasked with solving \(4x - 7 = 13\), your plan might involve isolating 'x' on one side. By executing the plan, adding 7 to both sides and then dividing by 4, you find \(x = 5\). Always look back to review and confirm that your solution is accurate and appropriate for the given problem.

Algebraic Thinking

Algebraic thinking is the cornerstone of all higher-level mathematical exploration. It's a sophisticated way of weaving patterns, functions, and symbols into a tapestry that represents quantities and their relationships. This form of reasoning transcends mere calculation; it's about understanding the 'why' as well as the 'how'.

Algebraic thinking encourages you to recognize and articulate the properties of numbers and operations, to see the symmetry and structure within mathematics, and to predict and analyze outcomes. By fostering this level of thinking, you're building a robust skill set that applies to a wide range of realistic scenarios, from determining the amount of paint needed for a room based on its area, to figuring out the best mobile phone plan for your needs based on usage and data. Encouraging this deep thinking enhances problem-solving skills and abstract reasoning, which are invaluable in both academic pursuits and real-world situations.