Problem 64 - Prealgebra

Question

Solve each equation. Check your solution. $$8 k=320$$

Step-by-Step Solution

Answer

The solution is \(k = 40\).

Step 1: Identify the equation

The given equation is \(8k = 320\). Our goal is to find the value of \(k\) that makes this equation true.

Step 2: Isolate the variable

To find \(k\), we need to isolate it on one side of the equation. We will do this by dividing both sides of the equation by 8. This gives us:\[ k = \frac{320}{8} \]

Step 3: Perform the division

Calculate \(\frac{320}{8}\) to find the value of \(k\). By dividing 320 by 8, we get:\(k = 40\).

Step 4: Check the solution

To ensure \(k = 40\) is correct, substitute \(k = 40\) back into the original equation:\(8 \times 40 = 320\).Doubling 40 twice yields 320, confirming our solution.

Key Concepts

Checking Solutions

After solving an equation, it's crucial to check if your solution is correct. This step confirms your understanding of the problem and ensures you didn't make any mistakes along the way.
Checking a solution involves substituting the value you found back into the original equation. If both sides of the equation remain equal after substitution, your solution is verified.
For example, with the equation \(8k = 320\) where we found \(k = 40\), substitute 40 back into the equation: \(8 \times 40 = 320\).
The left side equals the right side, confirming that \(k = 40\) is indeed the correct answer.

• Always remember to check your solution.
• This step helps to catch calculation errors.
• It reinforces your understanding of the concept.

Checking isn’t just a step but a reinforcement of learning.

Division in Algebra

Division is a fundamental skill in algebra used to simplify equations and solve for variables. It involves splitting a quantity into equal parts and is especially important for isolating variables.
In algebra, when you have an equation like \(8k = 320\), you can use division to solve for \(k\). You perform division by dividing both sides of the equation by the same number. This keeps the equation balanced.
Here is how it works: Divide both sides by 8, shown mathematically as \(k = \frac{320}{8}\).
Performing this division tells us \(k = 40\).

• Division helps undo multiplication.
• Ensures that we simplify the equation correctly.
• Maintains the balance of an equation by applying the operation equally to both sides.

Understanding how and why to perform division can lead to mastery in solving algebraic equations.

Isolation of Variables

Isolating a variable means getting the variable on one side of the equation by itself, a vital step in solving equations.
We perform operations like addition, subtraction, multiplication, or division on both sides of an equation to isolate the variable.
In \(8k = 320\), the goal is to isolate \(k\). By dividing both sides by 8, \(k\) stands alone as \(k = 40\).

• Isolating helps in finding what the variable specifically is.
• Each algebraic step should maintain the equality of the equation.
• Ensures a clear solution path by simplifying the problem.

This process forms the backbone of algebraic problem-solving because once the variable is isolated, you can see the solution clearly.