Problem 25 - Algebra for College Students

Question

Solve each equation. \(0.3(2 t+0.1)=8.43\)

Step-by-Step Solution

Answer

The solution to the equation is \(t = 14\).

Step 1: Distribute the Constant

To start, distribute the 0.3 to both terms inside the parentheses. This means multiplying 0.3 by each term:\(0.3 imes 2t + 0.3 imes 0.1 = 8.43\). This simplifies to:\(0.6t + 0.03 = 8.43\).

Step 2: Isolate the Variable Term

Next, subtract 0.03 from both sides to get the term containing \(t\) by itself:\(0.6t = 8.43 - 0.03\).This simplifies to:\(0.6t = 8.4\).

Step 3: Solve for t

Now, divide both sides by 0.6 to solve for \(t\):\(t = \frac{8.4}{0.6}\).This division yields:\(t = 14\).

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that helps in simplifying expressions. It allows us to "distribute" or multiply a single term across terms inside a parenthesis. In the given equation, we apply the distributive property to the expression \(0.3(2t + 0.1)\). This means we multiply 0.3 by each term inside the parentheses:

• 0.3 is multiplied by \(2t\), which results in \(0.6t\).
• 0.3 is also multiplied by 0.1, resulting in 0.03.

After applying the distributive property, our equation becomes \(0.6t + 0.03 = 8.43\).
Practicing the distributive property helps in breaking down more complex equations and makes it easier to manage multiple terms. It is particularly useful when equations involve parentheses, leading to simpler and often more straightforward expressions.

How to Isolate the Variable

Isolating the variable means arranging the equation so that the variable is on one side of the equation by itself. This helps in clearly seeing the value that the variable represents. In the expression \(0.6t + 0.03 = 8.43\), our goal is to have \(t\) by itself on one side.
The first step is to remove any constants from the side with the variable. We do this by subtracting 0.03 from both sides of the equation:

• On the left side: \(0.6t + 0.03 - 0.03 = 0.6t\).
• On the right side: \(8.43 - 0.03 = 8.4\).

Now, our equation simplifies to \(0.6t = 8.4\).
Isolating the variable ensures you're working with a more straightforward equation that is easier to solve. This step is crucial in making final calculations to find the value of unknowns in equations.

Solving Linear Equations

Solving a linear equation involves finding the value of the variable that makes the equation true. After isolating the variable, the next step is to solve for it explicitly. The current equation is \(0.6t = 8.4\).
To solve for \(t\), divide both sides by 0.6:

• The left side becomes: \(\frac{0.6t}{0.6} = t\).
• The right side becomes: \(\frac{8.4}{0.6} = 14\).

Thus, \(t = 14\) is the solution to the equation.
Solving linear equations typically involves operations such as addition, subtraction, multiplication, and division. These operations simplify the equation, allowing you to find the precise value of the variable. Mastering this skill aids in tackling more complex algebraic problems efficiently.