Problem 9 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check your solution in the original equation. $$ \frac{y}{4}=8 $$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{y}{4} = 8\) is \(y = 32\).

Step 1: Isolate the Variable y

Multiply both sides of the equation by 4 to cancel out the denominator on the left side. This results in: \(y = 4 \times 8\).

Step 2: Solve for y

After multiplying, it yields \(y = 32\). This is the solution for \(y\).

Step 3: Check the solution

The solution should then be checked by substituting it back into the original equation. If \(y = 32\), then the left side of the equation would be \(\frac{32}{4}\) which equals 8, thus confirming the solution is correct.

Key Concepts

Isolating the Variable

Understanding the process of isolating the variable is a fundamental step in algebra. It involves rearranging the equation so that the unknown variable stands alone on one side of the equation. In our example, \( y/4 = 8 \), the goal is to get \( y \) by itself. This is achieved by performing the same operation on both sides of the equation, ensuring that the equation remains balanced. Multiply both sides by 4 and you effectively remove the denominator:
\begin{align*}4 \times \frac{y}{4} &= 4 \times 8 \y &= 32\rend{align*}
The number 4 is the multiplicative inverse of \( \frac{1}{4} \) (the coefficient of \( y \) in our equation), which is why multiplication by 4 is chosen as the operation to isolate \( y \).

Equation Solving Steps

Solving algebraic equations typically follows a series of logical steps that lead to finding the value of the unknown variable. These equation solving steps are designed to simplify the equation to its most basic form. Let's break down the steps applied to our example equation:

• Multiply both sides by 4 to cancel out the \( \frac{1}{4} \) attached to \( y \).
• Perform the multiplication on the right side, which simplifies the equation to \( y = 32 \).

These steps led to identifying the value of \( y \). It is essential to conduct each step correctly and not skip any of them, as doing so can lead to incorrect answers. Always start with the simplest operation that can simplify the equation without complicating it further.

Checking Algebraic Solutions

Once a solution to an algebraic equation is found, it is crucial to check the solution to ensure its accuracy. To check our solution for the equation \( y/4 = 8 \), we substitute \( y \) with 32:
\begin{align*}\frac{32}{4} &= 8 \8 &= 8\rend{align*}
The equality holds true, confirming that the solution \( y = 32 \) is correct. This step acts as verification and can catch any arithmetic errors made during the solving process. If the check reveals a mismatch, it indicates a mistake has been made, necessitating a review of the previous steps to locate and correct the error.