Problem 24 - Prealgebra

Question

Solve each equation. Check your solution. $$k-4.1=-9.38$$

Step-by-Step Solution

Answer

The solution to the equation is \( k = -5.28 \).

Step 1: Isolate the Variable

To solve the equation \( k - 4.1 = -9.38 \), the first step is to isolate the variable \( k \). We can do this by adding 4.1 to both sides of the equation. This gives us:\[ k - 4.1 + 4.1 = -9.38 + 4.1 \]This simplifies to:\[ k = -9.38 + 4.1 \]

Step 2: Simplify the Equation

Now, we need to perform the addition on the right side of the equation:\[ -9.38 + 4.1 = -5.28 \]Thus, the value of \( k \) is:\[ k = -5.28 \]

Step 3: Check the Solution

To verify the solution, substitute \( k = -5.28 \) back into the original equation \( k - 4.1 = -9.38 \):\[ -5.28 - 4.1 = -9.38 \]Perform the subtraction on the left side:\[ -5.28 - 4.1 = -9.38 \]This confirms that both sides of the equation are equal, verifying our solution is correct.

Key Concepts

Isolating Variables

When solving equations, isolating the variable is the first key step. It involves rearranging the equation so that the variable you're trying to solve for stands alone on one side. To do this effectively, you often use the opposite operation of what's currently being applied to the variable. For example, in the equation \( k - 4.1 = -9.38 \), the operation affecting \( k \) is subtraction by 4.1. To isolate \( k \), you undo this subtraction by adding 4.1 to both sides of the equation:

• Add 4.1 to both sides: \( k - 4.1 + 4.1 = -9.38 + 4.1 \).
• This simplifies to \( k = -9.38 + 4.1 \).

Each equation is like a balance scale. By performing the same operation on both sides, you maintain balance and move closer to finding the value of the variable.

Checking Solutions

After finding a potential solution, it's important to check if it's correct by substituting the solution back into the original equation. This step ensures that no mistakes were made during calculations. Let's see how it's done for the example \( k = -5.28 \):

• Substitute the value back: \( k - 4.1 = -9.38 \) becomes \( -5.28 - 4.1 = -9.38 \).
• Perform the subtraction: \( -5.28 - 4.1 = -9.38 \).

Since both sides of the equation are equal, the solution is verified.
Correctly checking your work avoids errors and confirms your understanding of the problem-solving process.

Simplifying Equations

Simplifying equations is about performing arithmetic to make equations more manageable. This might involve combining like terms or performing operations to simplify expressions. In our example, once we isolate \( k \) in the equation \( k = -9.38 + 4.1 \), the next step is:

• Perform the addition: \( -9.38 + 4.1 = -5.28 \).

This operation gives the value of \( k \), making the equation more straightforward:
\( k = -5.28 \). Simplification reveals the solution in the clearest form and finalizes the result obtained by isolating the variable.