Problem 26 - Algebra A Combined Function

Question

Solve each equation. $$ \frac{6}{3 y}+\frac{3}{y}=1 $$

Step-by-Step Solution

Answer

The solution to the equation is \(y = 5\).

Step 1: Combine the Fractions

To combine \(\frac{6}{3y}\) and \(\frac{3}{y}\), note that their least common denominator is \(3y\). Rewrite each fraction with this common denominator: \(\frac{6}{3y}\) stays as it is, and \(\frac{3}{y}\) becomes \(\frac{9}{3y}\).

Step 2: Simplify the Combined Fraction

Combine the rewritten fractions: \(\frac{6}{3y} + \frac{9}{3y} = \frac{6 + 9}{3y} = \frac{15}{3y}\).

Step 3: Solve the Equation for y

Set the simplified fraction equal to 1: \(\frac{15}{3y} = 1\). Cross-multiply to solve for \(y\): \(15 = 3y\).

Step 4: Isolate y

Divide each side by 3: \(y = \frac{15}{3}\). Perform the division: \(y = 5\).

Step 5: Verify the Solution

Substitute \(y = 5\) back into the original equation to check: \(\frac{6}{3 \times 5} + \frac{3}{5} = \frac{2}{5} + \frac{3}{5} = 1\). The solution works, so \(y = 5\) is correct.

Key Concepts

Solving Equations

Solving equations is a fundamental aspect of algebra that you will encounter frequently. An equation is a mathematical statement that asserts the equality of two expressions. Solving an equation generally involves finding the value of an unknown variable that makes the equation true.
Here’s a simple approach to solving equations:

• Combine like terms on each side of the equation.
• Use algebraic operations, such as addition, subtraction, multiplication, or division, to simplify the equation.
• Isolate the variable to one side to determine its value.
• Check your solution by substituting it back into the original equation.

By following these steps, you can effectively solve basic to complex algebraic equations.

Fractions

Fractions represent parts of a whole and are written as one number over another, separated by a dividing line. In algebra, fractions often appear in equations and require particular attention. Understanding fractions is crucial because:

• They allow us to express non-whole numbers and ratios.
• Complex equations may require the manipulation of fractions.
• Finding a common denominator is necessary when adding or subtracting fractions.

When working with algebraic fractions, keep in mind the importance of maintaining equivalence by performing the same operation on both the numerator and the denominator.

Least Common Denominator

The least common denominator (LCD) is the smallest multiple that two or more denominators share. It is particularly useful when adding or subtracting fractions, as it provides a common basis for comparison.To determine the LCD:

• Identify the denominators of the fractions involved.
• Find common multiples of these denominators.
• Select the smallest of these multiples.

In our example,

• The denominators are \(3y\) and \(y\), so the LCD is \(3y\).

By rewriting each fraction with the LCD as the denominator, we simplify the process of combining and solving equations.

Isolate Variable

To isolate a variable means to manipulate an equation so that the variable appears alone on one side of the equation. This is a critical step in solving equations as it helps to identify the value of the unknown.Here's how to effectively isolate a variable:

• Perform inverse operations to "undo" any operations that involve the variable.
• For instance, if the variable is being multiplied by a number, divide both sides by that number.
• Similarly, add or subtract terms from both sides if necessary.

In the solution provided, \(15 = 3y\) is simplified by dividing each side by 3, leading to \(y = 5\). This step ensures that \(y\) is isolated, allowing us to easily see its value.