Problem 24 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ \frac{y}{3}=11 $$

Step-by-Step Solution

Answer

The solution is \( y = 33 \).

Step 1: Understand the equation

The given equation is a simple linear equation: \( \frac{y}{3} = 11 \). The goal is to find the value of \( y \) that makes this equation true.

Step 2: Isolate the variable

To solve for \( y \), we need to isolate it on one side of the equation. We can do this by multiplying both sides of the equation by 3 to eliminate the fraction: \[ 3 \cdot \frac{y}{3} = 11 \cdot 3 \] The left side of the equation simplifies to \( y \), while the right side becomes \( 33 \).

Step 3: Simplify the equation

Simplify the left side of the equation after applying the multiplication: \[ y = 33 \] This tells us that \( y = 33 \) is the solution to the equation.

Step 4: Verify the solution

To ensure the solution is correct, substitute \( y = 33 \) back into the original equation to check if it holds true: \[ \frac{33}{3} = 11 \] Simplifying the left side gives \( 11 \), which is equal to the right side, confirming that \( y = 33 \) is indeed the correct solution.

Key Concepts

Solving Equations

In mathematics, solving equations is a fundamental skill. It involves finding the unknown value that makes an equation true. When dealing with linear equations like \( \frac{y}{3} = 11 \), the goal is to solve for \( y \). Solving equations often follows a systematic approach, which typically includes understanding the equation, performing operations to isolate the variable, and verifying the solution.

Approaches to solving equations:

• Understand the type of equation you are working with.
• Identify what the equation is asking you to find—this is usually the variable.
• Use operations like addition, subtraction, multiplication, or division, to manipulate the equation and solve for the variable.

Comprehending the type and structure of an equation can guide you on the most appropriate method to use and, eventually, helps simplify the process of finding solutions.

Isolation of Variable

Isolation of variable is a crucial step in solving equations. It involves rearranging the equation so that the variable of interest stands alone on one side. In our example, \( \frac{y}{3} = 11 \), the variable is \( y \).

To isolate \( y \), we must eliminate the fraction by performing the inverse operation. Since \( y \) is divided by 3, the inverse operation is to multiply both sides of the equation by 3.

Steps for isolating the variable in \( \frac{y}{3} = 11 \):

• Multiply both sides by 3: \( 3 \times \frac{y}{3} = 3 \times 11 \).
• This simplifies to \( y = 33 \). The fraction is eliminated, leaving \( y \) isolated and equal to 33.

By successfully isolating the variable, we can determine its value, ensuring clarity and finding the exact solution to the equation.

Verification of Solution

Verification of solution is the final step to ensure your solution is correct. After solving an equation, it's important to check if the calculated value satisfies the original equation. This double-checking process verifies that no mistakes were made.

For the equation \( \frac{y}{3} = 11 \) with the solution \( y = 33 \), let's verify:

• Substitute \( y = 33 \) back into the original equation: \( \frac{33}{3} = 11 \).
• Simplify the left side: \( 33 \div 3 = 11 \).

Since the left side equals the right side (11), the solution \( y = 33 \) is confirmed to be correct.

Verification not only confirms the accuracy of your solution but also strengthens understanding and confidence in the problem-solving process.