Problem 59 - Elementary Algebra

Question

Solve each of the following equations and express the solutions in decimal form. Your calculator might be of some help. $$0.2(t+1.6)=3.4$$

Step-by-Step Solution

Answer

The solution is \( t = 15.4 \).

Step 1: Distribute Multiply

Start with the given equation: \[ 0.2(t + 1.6) = 3.4 \]Distribute the multiplication across the term in the parenthesis:\[ 0.2t + 0.2 imes 1.6 = 3.4 \]Calculate \(0.2 \times 1.6\):\[ 0.32 \]So the equation becomes:\[ 0.2t + 0.32 = 3.4 \]

Step 2: Isolate the Variable Term

Subtract \(0.32\) from both sides to isolate the term with the variable \(t\):\[ 0.2t + 0.32 - 0.32 = 3.4 - 0.32 \]This simplifies to:\[ 0.2t = 3.08 \]

Step 3: Solve for the Variable

Divide both sides by \(0.2\) to solve for \(t\).\[ \frac{0.2t}{0.2} = \frac{3.08}{0.2} \]This simplifies to:\[ t = 15.4 \]

Key Concepts

Decimal Solutions

When solving equations, especially those involving decimals, it's important to understand how to properly handle decimal numbers. In this exercise, we expressed the final solution as a decimal. Decimals are simply a way to represent fractions, such as 0.2 which represents the fraction \(\frac{2}{10}\).

When working with decimals, make sure to maintain precision.

• Calculate each step carefully to avoid rounding errors.
• Use a calculator where necessary to ensure accuracy.

Decimal solutions provide precise values, especially in science and engineering. Always double-check your results to ensure they align correctly with the context of the problem.

Distribution Property

The distribution property, sometimes called the distributive law, is a mathematical property used to simplify expressions by distributing a single term across terms within parentheses.

For example, in the equation \(0.2(t + 1.6) = 3.4\), the term \(0.2\) is distributed across \(t + 1.6\).
Here's the step-by-step breakdown:

• Multiply \(0.2\) by \(t\), giving \(0.2t\).
• Multiply \(0.2\) by \(1.6\), resulting in \(0.32\).

This turns the expression into \(0.2t + 0.32\).
The distributive property helps in simplifying and transforming equations so that they can easily be solved.

Isolating Variables

Isolating the variable means manipulating an equation so the unknown variable is on one side by itself, making it easier to solve. In our case, we wanted to isolate \(t\).

To do this, we subtracted \(0.32\) from both sides of the equation \(0.2t + 0.32 = 3.4\), resulting in \(0.2t = 3.08\).

Isolating variables involves using inverse operations. This step is crucial to simplify the equation so that you can find the solution for the unknown. Always ensure to:

• Perform the same operation on both sides of the equation equally.
• Double-check each step to ensure the variable is correctly isolated.

By isolating \(t\), the equation becomes much simpler.

Algebraic Manipulation

Algebraic manipulation involves rearranging algebraic expressions to solve for unknown variables. This includes using operations such as addition, subtraction, multiplication, and division.

In this exercise, after using the distributive property and isolating the variable \(t\), we divide both sides of the equation by \(0.2\). This gave us the value of \(t\):

• \(\frac{0.2t}{0.2} = t\)
• \(\frac{3.08}{0.2} = 15.4\)

The solution for \(t\) is therefore \(15.4\).
Algebraic manipulation is crucial in deciding the proper steps to solve equations and help in transforming complex expressions into simpler forms, leading you to the right answer.