Problem 101 - Introductory and Intermediate Algebra for College Students 4th

Question

I cleared the equation \(0.5 x+8.3=12.4\) of decimals by multiplying both sides by \(100 .\)

Step-by-Step Solution

Answer

The solution for x in the given equation is 8.2.

Step 1: Multiply Through by 100

First, multiply each term in the equation by 100. This gives \(50x + 830 = 1240.\)

Step 2: Isolate x

Next, subtract 830 from both sides of the equation to isolate x on one side. This gives \(50x = 410.\)

Step 3: Solve for x

Finally, divide both sides of the equation by 50 to solve for x. This yields \(x = 8.2.\)

Key Concepts

Decimal Elimination

Solving linear equations can sometimes involve cumbersome decimals which make calculations trickier. To simplify, we use decimal elimination. This technique allows us to work with whole numbers instead. In our example equation, \(0.5x + 8.3 = 12.4\), the decimals can be eliminated by multiplying every term by 100.

• Each term in the equation is multiplied by the same number, ensuring the balance of the equation is maintained.
• This transformation turns the equation into \(50x + 830 = 1240\), making it much easier to work with.

By eliminating decimals, we effectively clear the clutter and set ourselves up for more straightforward arithmetic operations. It's like decluttering your workspace before starting an important project!

Isolation of Variable

A key step in solving equations is isolating the variable, meaning to get it by itself on one side of the equation. This helps to determine the variable's true value. To isolate in the equation \(50x + 830 = 1240\), we need to manipulate the terms surrounding \(x\).

• Subtract 830 from both sides to remove it from the equation with \(x\). This gives \(50x = 410\).
• The subtraction step is crucial as it peels away layers around the variable, much like peeling an onion.

Once \(x\) is alone with its coefficient, we can easily find its value by tackling the next step. Isolating the variable systematically guides us to unravel the equation like a puzzle.

Equation Transformation

Equation transformation involves changing the form of an equation to make it easier to solve, without altering its solutions. It's a process of logical manipulation until we balance the equation in a way that is solvable. In the final step of our equation \(50x = 410\):

• Divide both sides by 50 to transform it into \(x = 8.2\).
• This step involves balancing the variables by performing the same operation on both sides, maintaining the equation's integrity.

Equation transformation is much like reshaping clay; you mold it into a form that reveals the answer effortlessly. Every step is aimed at homing in on the solution while preserving the equation's original truths.