Problem 33 - Prealgebra

Question

Solve each equation. Check your solution. $$0.4 d=2 d+1.24$$

Step-by-Step Solution

Answer

The solution is \(d = -0.775\).

Step 1: Move all terms to one side

Start by moving all terms involving the variable to one side of the equation. The equation is given as \(0.4d = 2d + 1.24\). Subtract \(2d\) from both sides to get \(0.4d - 2d = 1.24\).

Step 2: Simplify the expression

Simplify the left side of the equation. Combine the terms by subtracting: \(0.4d - 2d = -1.6d\), so the equation is now \(-1.6d = 1.24\).

Step 3: Solve for d

To solve for \(d\), divide both sides of the equation by \(-1.6\). This results in: \(d = \frac{1.24}{-1.6}\). Calculate the division to find \(d = -0.775\).

Step 4: Verify the solution

Substitute \(d = -0.775\) back into the original equation to verify. Compute both sides: \(0.4(-0.775) = -0.31\) and \(2(-0.775) + 1.24 = -1.55 + 1.24 = -0.31\). Both sides are equal, confirming the solution is correct.

Key Concepts

Equation Simplification

Simplifying an equation involves reducing it to its simplest form without changing its solution. Imagine an equation as a cluttered room you need to tidy up. The goal is to make it as simple as possible by moving terms around and combining like terms.

In the scenario where we have the equation \(0.4d = 2d + 1.24\), it is essential to have all terms involving the variable on one side. This helps to manage the equation more efficiently. To do this, subtract \(2d\) from both sides, yielding \(0.4d - 2d = 1.24\).

Your work is not over until you combine similar terms on the same side of the equation. For \(0.4d - 2d\), you would subtract the coefficients, which simplifies to \(-1.6d\). This makes the equation cleaner and easier to work with: \(-1.6d = 1.24\). This step is crucial for clarity and ease when solving further.

Variable Isolation

Isolating the variable essentially means getting the variable you are solving for by itself on one side of the equation. This process is like peeling away layers to expose the core. Once you've simplified the equation, your next goal is to "free" your variable from other numbers or coefficients around it.

In the equation \(-1.6d = 1.24\), to isolate \(d\), you need to perform the opposite operation of what is currently being done to \(d\). Since \(d\) is being multiplied by \(-1.6\), divide both sides by \(-1.6\) to get \(d = \frac{1.24}{-1.6}\).

Perform the division to find the value of \(d\), which in this case is \(d = -0.775\). The negative sign is crucial; it alters the direction of your solution on the number line, and incorrect handling can lead to incorrect outcomes.

Solution Verification

After finding a solution, it's always wise to verify its accuracy. This acts as a safety check and reassures you that you haven't made any mistakes during calculations. Think of it as testing a new gadget to make sure it works before taking it home.

Verification requires substituting the solution back into the original equation to ensure both sides equivalently balance. Let's substitute \(d = -0.775\) back into the equation \(0.4d = 2d + 1.24\). Calculate each side to verify:

• Left Side: \(0.4 \times -0.775 = -0.31\)
• Right Side: \(2 \times -0.775 + 1.24 = -1.55 + 1.24 = -0.31\)

The left and right sides of the equation equate, confirming that \(d = -0.775\) is indeed the correct solution. Verification is a crucial step, as errors can compound without it, compromising your results and understanding.