Problem 13 - Fundamentals of Mathematics

Question

Verify that each given value is a solution to the given equation. $$y-4=-6, y=-2$$

Step-by-Step Solution

Answer

Yes, \( y = -2 \) is a solution to the equation.

Step 1: Understand the Problem

We need to determine whether the value \( y = -2 \) satisfies the equation \( y - 4 = -6 \). This means we will substitute \( y = -2 \) into the equation and check if both sides are equal.

Step 2: Substitute the Value of y

Substitute \( y = -2 \) into the left-hand side of the equation: \( y - 4 \). This becomes:\(-2 - 4\).

Step 3: Perform the Arithmetic

Calculate \(-2 - 4\):\(-2 - 4 = -6\).

Step 4: Compare Both Sides of the Equation

After substituting and simplifying, we have \(-6\) on the left-hand side. The original equation is \( y - 4 = -6 \). Both sides of the equation are equal when \( y = -2 \).

Step 5: Conclude the Verification

Since substituting \( y = -2 \) into the equation results in both sides being equal, \( y = -2 \) is indeed a solution to the equation \( y - 4 = -6 \).

Key Concepts

Solution Verification

Verifying a solution to an equation is like checking if a lock properly fits with a key. After finding or assuming a value for a variable, we need to confirm that this value makes the equation balanced, meaning both sides of the equation are equal. Verification helps us ensure accuracy, much like checking an answer to a puzzle.
To verify a solution, follow these simple steps:

• Start by substituting the candidate value into the original equation.
• Simplify both sides, if necessary.
• Check if the resulting mathematical statement is true.

It's crucial because it gives confidence in the correctness of our solution.

Substitution Method

The substitution method is a straightforward technique used when solving equations. Think of it as "plug and play." Essentially, you replace a variable within an equation with a given number to see if it makes the equation true.
Here's how it works:

• Identify the variable in the equation you want to replace.
• Substitute the given or calculated value into the equation where the variable is.
• Complete any further calculation or simplification needed.

The substitution method is incredibly helpful because it allows us to test potential solutions efficiently. It's like taking a trial run to see if a hypothesis holds true.

Basic Arithmetic

Basic arithmetic involves fundamental mathematical operations: addition, subtraction, multiplication, and division. These operations are essential tools we use repeatedly when solving equations. Being proficient in basic arithmetic is necessary for simplifying equation steps effectively.
For example, in the process of substitution, you might encounter an arithmetic problem, like \(-2 - 4\). Knowing how to handle negative numbers and perform the operation gives you \(-6\). Keep practicing basic arithmetic for accuracy in solving more complex problems.

Equation Equality

Equation equality is the heart of verifying a solution. In an equation, both sides must be equal, maintaining a balance. It's like a scale in perfect balance - if one side changes, you need to adjust the other side equally to maintain balance.
When you substitute a value and simplify an equation, the goal is to check if both sides are equal. For instance, if after substitution you get both sides equaling \(-6\), then the equality holds.
Understanding how equation equality works ensures recognition of valid solutions and reinforces understanding of fundamental mathematics. Balancing equations is a vital skill in algebra and higher mathematics.