Problem 54 - Prealgebra

Question

Solve each equation. Check your solution. $$k-12=-40$$

Step-by-Step Solution

Answer

The solution to the equation is \(k = -28\).

Step 1: Isolate the Variable

We start by isolating the variable \(k\) on one side of the equation. The equation is \(k - 12 = -40\). We add 12 to both sides of this equation to get \(k = -40 + 12\).

Step 2: Simplify the Equation

Next, we simplify the right side of the equation. Calculate the sum: \(-40 + 12 = -28\). Thus, the equation simplifies to \(k = -28\).

Step 3: Check the Solution

To verify our solution, substitute \(k = -28\) back into the original equation \(k - 12 = -40\). Substitute to get \(-28 - 12 = -40\). Simplifying gives \(-40 = -40\), confirming our solution is correct.

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. These symbols represent numbers and can be used to express relationships and solve equations. In algebra, you often work with equations to find the value of an unknown variable, like in the equation \(k - 12 = -40\). Here, \(k\) represents the unknown value that we need to solve for. By understanding algebra, you can create equations that model real-world situations, predict outcomes, and solve problems efficiently. Practicing these skills helps build a foundation for higher-level math courses and enhance logical thinking.

Isolating Variables

Isolating variables means getting a variable by itself on one side of the equation. This is the main strategy in solving algebraic equations. For the equation \(k - 12 = -40\), our first goal was to isolate the variable \(k\). To do this, we performed the inverse operation to cancel out the \(-12\) attached to \(k\).

• Identify the operation affecting the variable. Here, it's subtraction ( - 12).
• To undo this, we add 12 to both sides of the equation ( + 12).

This step ensures that whatever you do to one side, you must also do to the other, maintaining the balance of the equation. Once \(k\) is isolated, you can proceed to find its value by simplification, as shown in the steps.

Checking Solutions

Once you have a solution, it is important to verify its correctness by checking the solution. This involves substituting the value back into the original equation to see if it holds true.
For our solution, \(k = -28\), plugging it back into the original equation \(-28 - 12 = -40\) allows you to ensure that both sides of the equation are still equal:

• Substitute: Replace \(k\) with -28.
• Simplify: Perform the operation to verify both sides are equal.
• Conclusion: If both sides match, the solution is correct.

Checking solutions helps in catching potential mistakes. It confirms the correctness of your answer and reveals any arithmetic errors unnoticed during solving.