Problem 108 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{6 y}{7}=5 $$

Step-by-Step Solution

Answer

The solution to the equation is \( y = \frac{35}{6} \).

Step 1: Understand the Equation

We are given the equation \( \frac{6y}{7} = 5 \). This is a simple linear equation where we need to find the value of \( y \).

Step 2: Clear the Fraction

To eliminate the fraction, we need to multiply both sides of the equation by 7. This gives us: \( 7 \times \frac{6y}{7} = 5 \times 7 \) This simplifies to: \( 6y = 35 \).

Step 3: Solve for y

Now that we have \( 6y = 35 \), we need to solve for \( y \) by dividing both sides by 6: \( y = \frac{35}{6} \).

Step 4: Simplify the Fraction

Check if \( \frac{35}{6} \) can be simplified further. Since 35 and 6 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Simplifying Fractions

Fractions are a way to represent numbers that aren't whole. Sometimes, fractions can be complex and may need to be simplified to make them easier to work with. Simplifying a fraction involves reducing it to its simplest form. A fraction is in its simplest form when the numerator (top number) and the denominator (bottom number) are as small as possible.

To simplify a fraction, you need to find the greatest common factor (GCF) of both the numerator and the denominator. This means finding the largest number that divides evenly into both. You then divide both the top and bottom of the fraction by this number.

### Key Points

• The fraction is simplified when no more common factors exist, other than 1.
• In the equation above, \( \frac{35}{6} \) is already simplified because 35 and 6 do not share any common factors other than 1.

Solving for a Variable

When solving an equation for a variable, you are essentially trying to isolate the variable on one side of the equation. This means you want the variable alone, without any coefficients or constants attached.

In our example, we start with the equation \( \frac{6y}{7} = 5 \). After clearing the fraction, we obtain \( 6y = 35 \). Here, the variable is \( y \), and our goal is to isolate \( y \) by performing the necessary algebraic operations.

### Steps to Solve for a Variable

• Perform operations that undo what has been done to the variable. In this case, \( y \) is multiplied by 6.
• To isolate \( y \), divide both sides of the equation by 6, resulting in \( y = \frac{35}{6} \).
• Always verify that your solution makes sense in the context of the problem.

Clearing Fractions

Fractions in equations can be inconvenient to work with, so one useful technique is to 'clear' them. Clearing fractions involves eliminating the fractions from an equation, usually by multiplying each term by a common denominator.

In our example, the equation is \( \frac{6y}{7} = 5 \). The fraction \( \frac{6y}{7} \) can make solving the equation more complex.

### How to Clear Fractions

• Identify the denominator of the fraction. Here, it is 7.
• Multiply every term in the equation by the denominator (7 in this case) to eliminate the fraction. This gives \( 6y = 35 \).
• Now the equation is free of fractions, making it easier to solve for \( y \).