Problem 54 - Prealgebra

Question

Solve each equation. Check your solution. $$5 y=60$$

Step-by-Step Solution

Answer

The solution to the equation is \( y = 12 \).

Step 1: Isolate the variable

To solve for \( y \), divide both sides of the equation by 5 to undo the multiplication. This isolates \( y \), giving: \[ y = \frac{60}{5} \]

Step 2: Perform the division

Calculate \( \frac{60}{5} \) to find the value of \( y \). So, you get: \[ y = 12 \]

Step 3: Check the solution

Substitute \( y = 12 \) back into the original equation to verify the solution: \[ 5 \times 12 = 60 \] Since both sides of the equation are equal, the solution is correct.

Key Concepts

Isolation of Variables

In equation solving, one of the most important steps is isolating the variable. This means making the variable stand alone on one side of the equation. When a variable is isolated, it helps us see the value it represents clearly and solves the equation.
To isolate a variable, you usually perform the inverse operation that is currently affecting the variable. For instance, if the variable is being multiplied by a number, the inverse operation would be division.
In our original exercise, the equation is given as \( 5y = 60 \). Here, \( y \) is multiplied by 5. To isolate \( y \), we need to divide both sides by 5. This undoes the multiplication and leaves us with \( y = \frac{60}{5} \).

• Remember: Whatever action you perform on one side, you must also do it to the other side to maintain balance.
• Isolating the variable helps in simplifying complex problems by reducing them step by step into a simpler form.

Isolating variables correctly is key for finding the solution to any equation efficiently.

Verification of Solutions

Solving an equation isn't complete without verifying the solutions. Verification is a critical step where you ensure that your obtained solution is accurate. It involves substituting the solution back into the original equation to check whether it satisfies the equation.
For example, in the exercise given, once \( y \) was calculated to be 12, we substitute \( y = 12 \) back into the original equation \( 5y = 60 \). When we replace \( y \) with 12, the equation becomes \( 5 \times 12 \), which equals 60. Since the left-hand side equals the right-hand side, the solution is verified.

• Verification is like a double-check; it catches errors and confirms correctness.
• Checking solutions also builds confidence and understanding of the entire problem-solving process.

Approaching each solution with a verification step ensures that your work is both precise and reliable.

Division in Algebra

Division in algebra is a fundamental concept often used to isolate variables in equations. When facing a multiplication equation, division serves as its counterpart to simplify the expression.
Consider the equation \( 5y = 60 \). Here, \( 5 \) is multiplied by \( y \). Division is used to undo this multiplication, allowing you to solve for \( y \). You divide both sides of the equation by 5: \[ y = \frac{60}{5} \]
This reduces to \( y = 12 \).

• Division helps in solving equations by breaking down complex expressions into manageable parts.
• Always ensure to perform the division equally on both sides to maintain balance in the equation.

Understanding division in algebra is crucial for tackling various equations and plays an essential role in isolating variables to find solutions effectively.