Problem 6 - Fundamentals of Mathematics

Question

Use the multiplication/division property of equality to solve each equation. Be sure to check each solution. $$ -k=-2 $$

Step-by-Step Solution

Answer

The solution is \(k = 2\).

Step 1: Identify the Equation

We begin with the equation \(-k = -2\)

Step 2: Apply the Multiplication Property of Equality

To solve for \(k\), we need to eliminate the negative sign in front of \(k\). This requires multiplying both sides of the equation by \(-1\), which will reverse the sign of both terms. Thus, \(-1 imes (-k) = -1 imes (-2)\).

Step 3: Simplify Expression

Multiplying by \(-1\), the equation simplifies to: \(k = 2\).

Step 4: Check the Solution

Substitute \(k = 2\) back into the original equation to verify:\(-k = -2\) becomes \(-2 = -2\), which is a true statement. Therefore, \(k = 2\) is the correct solution.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of a variable that makes an equation true. The core idea is to isolate the variable on one side of the equation, thus finding its solution.

In the provided exercise, the equation is \(-k = -2\). To isolate \(k\), we employ the **Multiplication Property of Equality**. This principle states that you can multiply both sides of an equation by the same non-zero number without altering the equality.

- The equation \(-k = -2\) can be solved by multiplying both sides by \(-1\).- This action is necessary to reverse the sign of the variable and solve for \(k\).
By multiplying both sides by \(-1\), we transform the equation into a simpler form: \(k = 2\).

Always aim to simplify the equation step by step until you have isolated the variable. This approach will make solving equations efficiently a routine process.

Checking Solutions

After solving an equation, it's essential to check if your solution is correct. This involves substituting the solution back into the original equation to verify that both sides equal.

For our equation \(-k = -2\), we found that \(k = 2\). Checking involves:

• Substitute \(k = 2\) back into the original equation.
• We have \(-2\) on the left and \(-2\) on the right, which confirms the correctness of our solved equation.

The purpose of checking solutions is to validate that the solution accurately satisfies the original equation. It's a reassurance step to prevent errors and confirm accuracy.

Negative Numbers

Negative numbers can sometimes confuse students, especially when they appear in equations. A negative number is any number less than zero and they follow certain rules when performing arithmetic operations.

**When Solving Equations:**- Multiplying or dividing both sides of an equation by a negative number reverses the sign of each term involved.
In our exercise, \(-k = -2\) is simplified by multiplying both sides by \(-1\) to cancel out the negatives.

**Key Tips to Remember:**

• Two negatives multiply to a positive: \(-1 \times -1 = 1\).
• Keeping track of negative signs is crucial; missing one can lead to incorrect solutions.
• Use the rules of negative numbers to confidently approach any equation involving them.

Understanding negative numbers and how they affect equations is important for solving more complex algebraic problems later on.