Problem 14 - Prealgebra

Question

Find the solution of each equation from the list given. $$14 k=42 ; 2,3,4$$

Step-by-Step Solution

Answer

The solution for the equation is \(k = 3\).

Step 1: Understanding the Equation

The given equation is \(14k = 42\). Our goal is to find the value of \(k\) that satisfies this equation.

Step 2: Isolate the Variable

To solve for \(k\), we need to isolate it on one side of the equation. This can be done by dividing both sides of the equation by 14, the coefficient of \(k\).

Step 3: Divide Both Sides by 14

When we divide both sides of the equation \(14k = 42\) by 14, we get \(k = \frac{42}{14}\). This simplifies to \(k = 3\).

Step 4: Verify the Solution

To ensure our solution is correct, substitute \(k = 3\) back into the original equation: \(14(3) = 42\). This results in \(42 = 42\), confirming that \(k = 3\) is indeed the correct solution.

Key Concepts

Isolate the Variable

When faced with solving linear equations, one of the first and most important steps is to isolate the variable. This allows us to find the solution to the equation.

In the equation given, \(14k = 42\), the variable to isolate is \(k\). Isolation means getting the variable on one side of the equation by itself. This is crucial because it reveals the value of the variable directly. To achieve isolation:

• Identify the operation and the coefficient attached to the variable. In this case, \(k\) is being multiplied by 14.
• Perform the inverse operation to remove the coefficient from the variable. Since multiplication by 14 is involved, its inverse operation is division by 14.

Once isolated, the equation becomes \(k = \frac{42}{14}\). This step is foundational and applies to almost any linear equation.

Verify the Solution

Verifying the solution of an equation is akin to a puzzle master checking that all pieces fit nicely. It ensures the accuracy and correctness of the derived solution. Let's examine the verification process:

• After isolating the variable and calculating \(k = 3\), substitute this value back into the original equation.
• The original equation is \(14k = 42\). By substituting \(k\) with 3, you calculate \(14 \times 3\).
• Ensure that the resulting equation holds true, meaning both sides equal each other.

In this case, \(14 \times 3 = 42\), which confirms the equation balances as \(42 = 42\). Verification is essential as it assures that the solution is not a mistake but the rightful answer.

Division in Equations

Understanding how to use division in equations is pivotal when solving for a variable. This operation is often used to eliminate coefficients attached to variables:

When we encounter coefficients, like the 14 in the equation \(14k = 42\), division is used to "undo" the multiplication between the coefficient and the variable. Here's how it works:

• Recognize the coefficient that needs to be removed to isolate the variable. In this exercise, it's the number 14.
• Divide both sides of the equation by this coefficient. Doing otherwise means the equation is no longer balanced. Here, dividing both sides by 14 simplifies the equation to \(k = \frac{42}{14}\).
• Perform the division to find the simplified solution of the variable. As calculated, \(k = 3\).

By mastering division in equations, you gain a strong tool for balancing equations and solving for unknowns efficiently.