Problem 79 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{-1}{6}=\frac{s}{-6} $$

Step-by-Step Solution

Answer

The solution to the equation is \( s = 1 \).

Step 1: Understand the Equation

We start with the equation \( \frac{-1}{6} = \frac{s}{-6} \). This is a simple proportion where both sides are fractions. To solve for \( s \), understand that we need to find the value of the variable \( s \) that makes the equation true.

Step 2: Cross-Multiply

Use the property of proportions that states if \( \frac{a}{b} = \frac{c}{d} \), then \( a \cdot d = b \cdot c \). For our equation \( \frac{-1}{6} = \frac{s}{-6} \), cross-multiply to get: \(-1 \times (-6) = 6 \times s\). This simplifies to: \(6 = 6s\).

Step 3: Solve for 's'

To isolate \( s \), divide both sides of the equation \( 6 = 6s \) by 6. This gives: \[ s = \frac{6}{6} \] Simplifying, we find: \( s = 1 \).

Key Concepts

Cross Multiplication

Cross multiplication is a useful technique when dealing with equations involving fractions or ratios. It's the process of multiplying the numerator of one fraction by the denominator of the other, across the equal sign. In essence, cross multiplication helps us eliminate the fractions in the equation, simplifying the problem to a basic linear equation.
For example, in the problem \( \frac{-1}{6} = \frac{s}{-6} \), you can set up your cross multiplication. This means multiplying \( -1 \) by \( -6 \) and \( s \) by \( 6 \). Thus, you will end up with the equation \(-1 \times (-6) = 6 \times s\), which simplifies to \(6 = 6s\).
Some key points to remember during cross multiplication:

• It can only be used when you have one fraction equal to another (proportion).
• Always perform the multiplication across the equals sign, ensuring accuracy.
• After multiplication, simplify the resulting equation to solve for the unknown variable.

Understanding cross multiplication is vital for solving proportional equations efficiently.

Isolation of Variables

Isolation of variables is the step we take to solve for an unknown in an equation. After performing cross multiplication, you typically have a simplified equation with the variable inside. The goal is to have the variable by itself on one side of the equation and the known value on the other.
After cross multiplying in our problem, we arrive at \(6 = 6s\). To isolate \(s\), we need to get rid of the 6 that is with \(s\). You can achieve this by performing the opposite operation, which is division in this case, since \(6\times s\) implies multiplication. Therefore, divide both sides by 6:
\[ s = \frac{6}{6} \]
This results in \(s = 1\).
Consider these points when isolating variables:

• Always perform the same operation on both sides of the equation to maintain equality.
• Use inverse operations to simplify. For instance, if a variable is multiplied by a number, divide by that number.
• Check your solution by substituting back to ensure it satisfies the original equation.

Proper isolation of variables leads directly to finding solutions for equations.

Fraction Simplification

Fraction simplification is a critical skill not only in solving equations but also in arithmetic. It involves reducing fractions to their simplest form, making them easier to work with. Simplifying fractions requires dividing the numerator and the denominator by their greatest common divisor (GCD).
In the example, once you have isolated \(s\) as \(\frac{6}{6}\), simplification is straightforward. Both numbers, the numerator and the denominator, are divided by 6, which is their GCD. So, \(\frac{6}{6} = 1\).
Key points on fraction simplification:

• Simplification ensures results are in the most compact form.
• Identify the greatest common divisor (GCD) of numbers to divide both the numerator and denominator.
• Always simplify as a final step when working with equations involving fractions.

Simplifying fractions not only makes results neater but also confirms that calculations are accurate for presenting solutions.