Problem 10 - Prealgebra

Question

Use the multiplication property of equality to solve each of the following equations. In each case, show all the steps. $$\frac{2}{3} y=18$$

Step-by-Step Solution

Answer

The solution is \( y = 27 \).

Step 1: Identify the Equation

Begin with the given equation: \( \frac{2}{3} y = 18 \). The goal is to solve for \( y \), which is the variable of interest.

Step 2: Understand the Multiplication Property of Equality

The multiplication property of equality states that you can multiply both sides of an equation by the same non-zero number without changing the equation's solution. Here, we want to eliminate the fraction.

Step 3: Multiply Both Sides by the Reciprocal

To isolate \( y \), we multiply both sides of the equation by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \). Thus, the equation becomes: \[ \frac{3}{2} \times \frac{2}{3} y = \frac{3}{2} \times 18 \].

Step 4: Simplify the Left Side

On the left side, multiplying \( \frac{3}{2} \times \frac{2}{3} \) results in 1 (since \( \frac{3}{2} \times \frac{2}{3} = 1 \)), leaving us with \( y \).

Step 5: Simplify the Right Side

On the right side, calculate \( \frac{3}{2} \times 18 \). First, multiply 18 by 3, getting 54, and then divide by 2, resulting in 27. Therefore, \( y = 27 \).

Step 6: Conclusion: Verification

Substitute \( y = 27 \) back into the original equation to verify. \( \frac{2}{3} \times 27 = 18 \). Simplifying the left side gives: \( 18 = 18 \), confirming \( y = 27 \) is correct.

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra that involves finding the value of an unknown variable that makes an equation true. Equations are like balanced scales, where each side must represent the same value. In solving an equation, the goal is to isolate the variable, usually represented by a letter like \( y \) or \( x \), on one side of the equation. This process involves performing operations such as addition, subtraction, multiplication, or division on both sides to preserve balance.

• Identify the variable you need to solve for.
• Use appropriate operations to undo any addition, subtraction, multiplication, or division affecting the variable.
• Check your solution by substituting it back into the original equation.

Solving equations helps to find the correct value for the unknown that resolves the equation, ensuring it holds true under the given conditions.

Fractions in Equations

When dealing with fractions in equations, the primary challenge is often how to eliminate the fraction to simplify the problem. Fractions can complicate equations because they introduce additional steps compared to whole numbers. However, with the multiplication property of equality, these challenges become manageable.
To tackle fractions:

• Identify the fraction you need to clear from the equation.
• Multiply both sides of the equation by the reciprocal of the fraction. The reciprocal of a fraction like \( \frac{a}{b} \) is \( \frac{b}{a} \).
• This multiplication cancels out the fraction, leaving the variable free to solve like a regular equation.

The process of clearing fractions by using their reciprocals ensures that the equation remains balanced while simplifying the solving process. It allows us to convert a complex-looking equation into a simpler form, facilitating the isolation and resolution of the variable.

Step-by-Step Solution

A step-by-step solution is a methodical approach to solving mathematical equations, breaking down the process into manageable stages. This way of solving makes it easier for learners to understand each part of the process and how they contribute to the overall solution.
In the context of an equation like \( \frac{2}{3}y = 18 \):

• First, clearly identify the equation and understand your goal (e.g., solving for \( y \)).
• Next, apply the multiplication property of equality to both sides, using the reciprocal of the existing fraction to simplify.
• Simplify each side step by step, ensuring that calculations are accurate.
• Once simplified, arrive at the solution for the variable.
• As a final step, substitute the solution back into the original equation to verify its correctness.

A step-by-step solution not only provides clarity but also reinforces understanding by sequentially demonstrating how each algebraic operation contributes to isolating and solving for the variable. This clarity is particularly helpful when learning how to handle more complex equations involving fractions.