Problem 60 - Elementary Algebra

Question

Solve each of the following equations and express the solutions in decimal form. Your calculator might be of some help. $$0.4(t-3.8)=2.2$$

Step-by-Step Solution

Answer

t = 9.3

Step 1: Distribute the Coefficient

First, distribute 0.4 to each term inside the parenthesis. This means multiplying 0.4 by both t and -3.8: \(0.4 \cdot t - 0.4 \cdot 3.8 = 0.4t - 1.52\).Now our equation is: \(0.4t - 1.52 = 2.2\).

Step 2: Isolate the Variable Term

Add 1.52 to both sides of the equation to move the constant term to the right side:\(0.4t - 1.52 + 1.52 = 2.2 + 1.52\).This simplifies to:\(0.4t = 3.72\).

Step 3: Solve for the Variable

Divide both sides by the coefficient of t, which is 0.4, to solve for t:\(\frac{0.4t}{0.4} = \frac{3.72}{0.4}\).This results in:\(t = 9.3\).

Key Concepts

Step-by-Step Solutions

Understanding step-by-step solutions is essential in solving linear equations or any math problem. The idea is to break down the solution process into manageable pieces, making it easier to follow and comprehend. Here's why step-by-step solutions matter:

• Clarity: Each step focuses on a small chunk of the problem, simplifying complex equations.
• Elimination of Errors: Following a methodical approach reduces the chances of overlooking mistakes.
• Learning Tool: Each step builds upon the previous one, reinforcing learning and understanding.

In our example, the solution process involved distributing a number, isolating the variable, and solving the equation. As you can see, breaking it down into these steps makes it more approachable, as each phase unfolded logically and progressively led to the final solution.

Decimal Form

Expressing solutions in decimal form is a crucial skill in mathematics. Decimals are a standard way to represent non-integer numbers, making them easy to read and understand. Here's how working with decimal forms can help you:

• Precision: Decimals provide more precise results compared to fractions.
• Universality: Many people and calculators use decimal systems, making it a universal way to present numbers.
• Comparison: It's often easier to compare numbers when they're expressed in decimal form rather than fractions.

In our exercise, the final solution is presented as a decimal (9.3). Decimals allow you to clearly see the solution's magnitude and make further calculations more straightforward if needed.

Distributive Property

The distributive property is one of the foundational concepts in algebra that helps simplify equations. It states that a single term multiplied by a sum inside a parenthesis can be distributed to each term. This is important because it simplifies equations, allowing for easier manipulation. Here's how it works:

• Distribution: Multiply the term outside the parenthesis by each inside, e.g., \(a(b + c) = ab + ac\).
• Simplification: Break down complex expressions into simpler parts.
• Mental Math: Enhances quick calculations and improves number sense.

In the problem solved, the distributive property was used in the first step to transform \(0.4(t - 3.8)\) into \(0.4t - 1.52\). Understanding this property can significantly improve one’s ability to solve linear equations efficiently.