Problem 25 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ \frac{-z}{6}=-14 $$

Step-by-Step Solution

Answer

The solution is \(z = 84\).

Step 1: Eliminate the Fraction

Start by eliminating the fraction in the equation \( \frac{-z}{6} = -14 \). We can do this by multiplying both sides of the equation by 6 to get rid of the denominator.This gives us:\[ -z = -14 \times 6 \]

Step 2: Simplify the Right Side

Next, simplify the right side of the equation:\[ -z = -84 \]So, multiplying \(-14\) by \(6\) gives \(-84\).

Step 3: Solve for \(z\)

Since the equation \(-z = -84\) shows \(z\) with a negative sign, we need to multiply by \(-1\) to isolate \(z\):\[ z = 84 \]

Step 4: Check the Solution

Verify the solution by substituting \(z = 84\) back into the original equation:\[ \frac{-84}{6} = -14 \]Calculate the left side:\[ -14 = -14 \]Since both sides are equal, the solution is verified.

Key Concepts

Understanding Fractions in Equations

Fractions in equations can be intimidating, but they can be simplified. When you have a fraction with a variable in the numerator, like \( \frac{-z}{6} = -14 \), your first instinct should be to clear the fraction for easier manipulation. To remove the fraction, we multiply both sides of the equation by the denominator.

• This action essentially "undoes" the division that the fraction represents.
• In our example, multiplying both sides by 6 eliminates the fraction entirely.

After doing this, you're left with a simpler equation without fractions. This makes the process of solving for the variable more straightforward and less prone to mistakes due to fraction handling.

Isolating Variables in Equations

Once the fraction is gone, the next goal is to isolate the variable. In our exercise, we ended up with the equation \(-z = -84\) after eliminating the fraction. Notice the negative sign in front of the variable.

• In such cases, we can multiply both sides of the equation by \(-1\) to make the variable positive.

This step is crucial because it simplifies the equation, revealing the positive value of \(z\) directly. For a quick recap:

• We had \(-z = -84\).
• By multiplying by \(-1\), we got \(z = 84\).

Isolating variables often requires reversing operations like subtraction or division, ensuring you're solving for the variable correctly.

Checking Solutions for Accuracy

Always remember to check your solution to confirm its accuracy. This is a fundamental step in solving equations because it verifies your work. In our example, we found \(z = 84\). Now, substitute this value back into the original equation:

• Start by replacing \(z\) with 84 in the equation \( \frac{-z}{6} = -14\).
• The left side becomes \( \frac{-84}{6} \).
• Calculate \( \frac{-84}{6} \) to see if it equals \(-14\).

By doing the arithmetic, we indeed find that \( \frac{-84}{6} = -14\), confirming the equality. This step ensures you've reached the correct solution, reinforcing the importance of double-checking your work in mathematics.