Problem 40 - Basic College Mathematics

Question

Solve. \(y \cdot \frac{5}{6}=120\)

Step-by-Step Solution

Answer

The value of \(y\) is 144.

Step 1: Write the Original Equation

Start with the original equation given in the problem: \[y \times \frac{5}{6} = 120\]

Step 2: Multiply Both Sides by the Reciprocal

To isolate \(y\), multiply both sides of the equation by the reciprocal of \(\frac{5}{6}\), which is \(\frac{6}{5}\): \[ \left( y \times \frac{5}{6} \right) \times \frac{6}{5} = 120 \times \frac{6}{5} \]

Step 3: Simplify the Equation

On the left side of the equation, the \(\frac{5}{6}\) and the \(\frac{6}{5}\) cancel each other out, leaving \(y\). On the right side, perform the multiplication: \[y = 120 \times \frac{6}{5}\] \[y = 120 \times 1.2 = 144\]

Step 4: Write the Final Solution

The value of \(y\) that satisfies the equation is \(y = 144\).

Key Concepts

isolating variables

When solving linear equations, isolating the variable is often the first step. This means getting the variable by itself on one side of the equation. In our example, the original equation is \( y \times \frac{5}{6} = 120. \) To isolate \( y \), we must remove \( \frac{5}{6} \) from the left side. We do this by performing the opposite operation: multiplication by the reciprocal of \( \frac{5}{6} \).

• Identify the operation affecting the variable
• Apply the inverse (or opposite) operation to both sides of the equation

By focusing on isolating the variable, you can simplify the equation step by step.

multiplying reciprocals

The reciprocal of a number is essentially \( \frac{1}{\text{number}}. \) For example, the reciprocal of \( \frac{5}{6} \) is \( \frac{6}{5}.\) Multiplying a number by its reciprocal results in 1. In our specific equation \( y \times \frac{5}{6} \times \frac{6}{5} = 120 \times \frac{6}{5}, \) the reciprocals cancel out:

• \( y \times \frac{5}{6} \times \frac{6}{5} = y \times 1 = y\)
• \( 120 \times \frac{6}{5} = 144\)

This makes solving for \( y \)

simplifying equations

Simplifying equations is all about making them easier to understand and solve. In our equation, once we multiply both sides by \( \frac{6}{5}, \) we simplify as follows:\( y = 120 \times \frac{6}{5} \)

• Use basic arithmetic to simplify: \( 120 \times \frac{6}{5} = 120 \times 1.2 = 144 \)
• Make sure to perform the same operations on both sides.

Simplification often involves breaking down the steps and performing operations systematically.

basic algebra

Basic algebra includes operations involving variables and constants using the four fundamental arithmetic operations: addition, subtraction, multiplication, and division. In this problem, we're dealing with multiplication and division to isolate and solve for a variable. Here is a quick breakdown:

1. Identify terms and constants: The equation provided is \( y \times \frac{5}{6} = 120\).

• Variables are symbols that represent unknown values (e.g., \( y \)).
• Constants are known values (e.g., 120).

Apply basic operations to balance and solve: Perform the same operations on both sides to maintain equality.

Understanding these key ideas helps solve algebra problems efficiently.