Problem 15 - Fundamentals of Mathematics

Question

Verify that each given value is a solution to the given equation. $$5 y+6=-14, y=-4$$

Step-by-Step Solution

Answer

Yes, \( y = -4 \) is a solution.

Step 1: Substitute the Value

Start by substituting the given value of the variable into the equation. Here we have the equation \( 5y + 6 = -14 \), and we're given \( y = -4 \). Substituting \( y = -4 \) into the equation gives: \( 5(-4) + 6 = -14 \).

Step 2: Simplify the Equation

Now, simplify the left side of the equation. First, multiply \( 5 \) by \( -4 \) which results in \( -20 \). So the equation becomes: \( -20 + 6 = -14 \).

Step 3: Combine Like Terms

Combine the terms on the left side: \( -20 + 6 \). This results in \( -14 \). So the equation now reads: \( -14 = -14 \).

Step 4: Verify the Result

The left side of the equation \( -14 \) matches the right side \( -14 \), confirming that \( y = -4 \) is indeed a solution to the equation \( 5y + 6 = -14 \).

Key Concepts

Substitution Method

The substitution method is a powerful technique, often used for solving equations. This method involves replacing a variable in an equation with a given value. It helps to simplify both sides of the equation, allowing you to verify solutions with more ease. Let's break it down simply:

• Identify the variable and the value you are going to substitute.
• Replace the variable in the equation with the given value.
• Simplify the equation, reducing it to a basic true/false statement that helps you verify if the substitution was correct.

Remember, substitution is not limited to finding solutions, but it also plays an important role in working with systems of equations. It's invaluable in both individual math problems and higher-level mathematics.

Linear Equations

Linear equations are fundamental in mathematics, often taking the form of a straight line when graphed. These equations typically involve variables raised only to the first power. Characteristics of linear equations include:

• The structure often includes one variable or multiple variables summed together.
• The general form, like 'ax + b = c'.
• When graphed, they form a straight line, hence their name.

An effective way to understand linear equations is by analyzing both sides of the equation to see how they balance. For example, in the equation \(5y + 6 = -14\), simplifying either side helps in checking the validity of solutions like \(y = -4\). Understanding this balancing act is critical in honing your math problem-solving skills.

Mathematical Reasoning

Mathematical reasoning refers to the logical process of deducing the correct solution or conclusion from the given premises. It involves critical thinking and the ability to follow a coherent path from problem to solution. With mathematical reasoning, you:

• Analyze the problem and understand what it entails.
• Use logic to simplify and solve equations.
• Verify that your solution is correct by substituting back into the original problem.

An example of mathematical reasoning is proving an equation's solution through the substitution method. Each step must make logical sense in progressing towards verifying that the solution is valid. This builds confidence in your solution, ensuring it stands up to scrutiny and fulfills the mathematical query posed by the problem.