Problem 16 - Algebra for College Students

Question

Solve each equation. \(0.13 t-3.4=0.08 t-0.4\)

Step-by-Step Solution

Answer

The solution to the equation is \(t = 60\).

Step 1: Isolate Variable Terms

First, get all the terms with the variable on one side of the equation. Subtract \(0.08t\) from both sides of the equation:\[0.13t - 3.4 - 0.08t = -0.4\] which simplifies to:\[0.05t - 3.4 = -0.4\]

Step 2: Eliminate Constant Term

Add \(3.4\) to both sides to eliminate the constant term on the left side:\[0.05t = 3.0\]

Step 3: Solve for the Variable

Divide both sides by \(0.05\) to solve for \(t\):\[t = \frac{3.0}{0.05}\] which simplifies to:\[t = 60\]

Key Concepts

Isolating Variables

When solving a linear equation, the first crucial step is isolating the variable. This means getting the variable, often represented as "x" or "t", all by itself on one side of the equation. To isolate the variable, look at the original equation:\[0.13t - 3.4 = 0.08t - 0.4\]Notice two terms contain "t". By bringing them together on one side, you make the equation easier to solve. Here, we subtract \(0.08t\) from both sides:\[0.13t - 0.08t - 3.4 = -0.4\]This simplifies to:\[0.05t - 3.4 = -0.4\]Why do we subtract? By performing the opposite operation (subtraction of \(0.08t\)), we effectively eliminate the "t" term from the right side. This leaves only one variable term on the left. Remember, the goal is to gradually focus the equation onto the variable without distractions from constant or additional variable terms.

Solving Equations Step by Step

Taking it step by step makes solving equations more manageable. Let’s guide you through it with the simplified equation:\[0.05t - 3.4 = -0.4\]It's crucial to tackle one part of the equation at a time. First, ensure all the terms containing variables are on one side. Once isolated, focus on isolating the variable completely by removing other number terms around it. Doing it step by step minimizes mistakes and clarifies every action with the goal in mind.After addressing the variable, eliminate any constants nearby. This means if there's a subtraction or addition of a number with the variable, remedy it through addition or subtraction. Keep your work organized:

• First, address variable terms so they sit together.
• Next, deal with constant terms one at a time.
• Simplify equations progressively until the variable stands alone.

Eliminating Constant Terms

Finally, let's discuss eliminating constant terms. Once your variables have come together, constants might still surround them. To isolate the variable fully, you need to "clear" these constant terms.Taking the equation from earlier:\[0.05t - 3.4 = -0.4\]You need to remove \(-3.4\) from the left side. To do this, add \(3.4\) to both sides:\[0.05t - 3.4 + 3.4 = -0.4 + 3.4\]This simplifies to:\[0.05t = 3.0\]Adding the same number to both sides leaves the balance of the equation intact. It’s key to perform all operations equally across the equation to maintain balance. Constant terms are numbers detached from variables. Removing them is like peeling an onion, bringing you closer to the core variable for a solution. Once constants are eliminated, solving becomes straightforward. Divide or multiply as needed to finally isolate your variable.