Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$\frac{3}{y}+\frac{6}{y}-\frac{1}{y}=11$$

Step-by-Step Solution

Answer

The solution is \(y = \frac{8}{11}\).

Step 1: Simplify the Left Side of the Equation

Combine the fractions on the left-hand side of the equation. Since all fractions have the same denominator "y", you can simply add and subtract their numerators: \(\frac{3}{y} + \frac{6}{y} - \frac{1}{y} = \frac{3 + 6 - 1}{y} = \frac{8}{y}\).

Step 2: Rewrite the Equation

The equation now becomes \(\frac{8}{y} = 11\).

Step 3: Solve for y

Eliminate the fraction by multiplying both sides of the equation by \(y\) to get \(8 = 11y\).

Step 4: Isolate y

Divide both sides of the equation by 11 to solve for \(y\): \(y = \frac{8}{11}\).

Key Concepts

Combining Like Terms

When solving rational equations, combining like terms is a crucial step. It simplifies the equation by merging similar elements, making it easier to solve. In the given exercise, the fractions have the same denominator which means they can be directly combined by adding or subtracting their numerators. Consider fractions \(\frac{3}{y}, \frac{6}{y}, \frac{-1}{y}\). Since they all share the denominator \(y\), you combine their numerators simply: \(3 + 6 - 1 = 8\). So the expression becomes \(\frac{8}{y}\). Here are the steps summed up:

• Identify fractions with the same denominator.
• Add or subtract the numerators as indicated.
• Place the result over the common denominator.

Combining these terms will help clarify the equation for the next steps in solving.

Fraction Operations

When dealing with rational equations such as \(\frac{8}{y} = 11\), special attention should be paid to handling fractions. Knowing fraction operations is essential to simplify and solve the equation efficiently. In this exercise, once the left side is simplified to \(\frac{8}{y}\), you are ready to act on the whole equation.A critical operation here is eliminating the fraction to make the equation easier to handle. To do this, multiply both sides by the denominator, which in this case is \(y\). This step clears the denominator:

• \(\frac{8}{y} \times y = 8\)
• \(11 \times y = 11y\)

The new equation \(8 = 11y\) is a much simpler non-fractional equation, allowing you to proceed to the next steps with ease.

Isolating Variables

Once you have a simplified equation, the next major step is to isolate the variable you are solving for, like \(y\) in the equation \(8 = 11y\). Isolation involves finding a way to express the variable as the subject of the formula.For the equation \(8 = 11y\), isolate \(y\) by dividing both sides by 11: \(y = \frac{8}{11}\). This operation ensures that \(y\) is now by itself on one side, giving you the solution.Key points to remember when isolating variables:

• Perform the same operation on both sides to maintain equality.
• Simplify the equation step-by-step until the variable is alone.
• Ensure all calculations are accurate, especially with fractions involved.

This methodical approach helps you find the correct solution and understand the logic behind the steps.