Problem 52 - Algebra A Combined Function

Question

Solve. $$ \frac{1}{11}=y+\frac{10}{11} $$

Step-by-Step Solution

Answer

The solution is \( y = \frac{-9}{11} \).

Step 1: Understand the Equation

The given equation is \( \frac{1}{11} = y + \frac{10}{11} \). It has a rational number on one side and \( y \), which is the variable we need to solve for, on the other. Both sides involve fractions with a common denominator.

Step 2: Simplify the Equation

To isolate \( y \), subtract \( \frac{10}{11} \) from both sides of the equation: \[ \frac{1}{11} - \frac{10}{11} = y \].

Step 3: Perform the Subtraction

Since both fractions have the same denominator, subtract the numerators: \( 1 - 10 = -9 \). This gives \[ \frac{-9}{11} = y \].

Step 4: Conclusion

We have isolated \( y \) on one side of the equation, giving the solution \( y = \frac{-9}{11} \).

Key Concepts

Fractions

When working with fractions, think of them as parts of a whole. If you divide something into equal parts, a fraction represents how many of those parts you have.
For instance, in the fraction \( \frac{1}{11} \), the number 1 is the numerator, showing how many parts are considered, while 11 is the denominator, indicating into how many parts the whole is divided.
Understanding the roles of numerators and denominators is key to any operation with fractions, such as addition or subtraction.

• Ensure both fractions have the same denominator before you add or subtract them.
• If they don’t, you’ll need to find a common denominator first.
• Sometimes, fractions can be simplified if the numerator and denominator have a common factor.

These concepts form the foundation for solving equations involving fractions by allowing us to handle parts of numbers in a logical way.

Isolating Variables

In algebra, solving an equation means finding out what value the variable represents. To achieve this, we often need to isolate the variable, meaning we get it alone on one side of the equation.
Let's take the equation \( \frac{1}{11} = y + \frac{10}{11} \). To isolate \( y \), we need to perform operations that will leave \( y \) by itself on one side.
Here are some tips for isolating variables:

• Do the same operation on both sides of the equation, like adding, subtracting, multiplying, or dividing both sides by the same number.
• Focus on eliminating additional terms or coefficients attached to the variable. In our case, we subtract \( \frac{10}{11} \) from both sides to eliminate it.

When you successfully isolate the variable, you can clearly see its value and thus solve the equation.

Subtracting Fractions

Subtracting fractions is straightforward if you understand that you need common denominators. In the equation \( \frac{1}{11} = y + \frac{10}{11} \), both fractions have the same denominator (11), which simplifies the subtraction process.
Here's how you subtract fractions:

• Ensure both fractions have the same denominator.
• Subtract the numerators, which are the upper numbers of the fractions.
• Keep the denominator the same.
• Simplify the fraction if possible.

For example, subtracting \( \frac{10}{11} \) from \( \frac{1}{11} \) involves subtracting 10 from 1 while keeping the denominator 11, resulting in \( \frac{-9}{11} \).
Knowing how to subtract fractions efficiently helps in solving equations where fractions appear, leading to quicker and more accurate solutions.