Problem 17 - Elementary Algebra

Question

Solve each of the equations. $$0.09 x+0.1(2 x)=130.5$$

Step-by-Step Solution

Answer

The solution is \(x \approx 450\).

Step 1: Distribute the 0.1 to the terms inside the parenthesis

First, apply the distributive property to the term \(0.1(2x)\). This means you'll multiply \(0.1\) by \(2x\), which results in \(0.2x\). So, the equation becomes: \[0.09x + 0.2x = 130.5\].

Step 2: Combine like terms

Next, combine the \(x\)-terms on the left side of the equation. Add \(0.09x\) and \(0.2x\) together to get \(0.29x\). Thus, the equation simplifies to: \[0.29x = 130.5\].

Step 3: Solve for x

To isolate \(x\), divide both sides of the equation by \(0.29\). This gives: \[x = \frac{130.5}{0.29}\]. Calculate this division to find the value of \(x\).

Step 4: Simplify the division

Perform the division \(\frac{130.5}{0.29}\). By calculating, you find that \(x \approx 450\).

Key Concepts

Distributive Property

Let's start by understanding the distributive property, a key concept in algebra that helps simplify expressions. This property allows you to multiply a single term by each term inside a set of parentheses. It might sound a bit tricky, but it's quite straightforward with an example.
For instance, if you have the expression \(a(b + c)\), you use the distributive property to spread out the multiplication: first multiply \(a\) by \(b\) and then \(a\) by \(c\). This results in \(ab + ac\).

In our original exercise, the property was used to handle \(0.1(2x)\). By distributing \(0.1\), we multiplied it by \(2x\) to get \(0.2x\). This step is essential because it eliminates the parenthesis, making the equation easier to solve.

Using the distributive property is a fundamental skill in algebra. It sets the stage for simplifying equations effectively.

Combining Like Terms

Once you have used the distributive property, the next step is to combine like terms. This concept refers to the process of simplifying expressions by adding or subtracting terms that have the same variable part.
If an equation contains multiple terms with \(x\), for example, these can often be added together to simplify the problem.

In our exercise, we initially had \(0.09x + 0.2x\). Both \(0.09x\) and \(0.2x\) are like terms because they contain the variable \('x'\). Therefore, we can combine them by simply adding the coefficients: \(0.09 + 0.2 = 0.29\).
This gives us a new equation: \(0.29x = 130.5\).

By combining like terms, we streamline the equation, which makes solving it much more straightforward. It’s all about looking for terms that have the same variable so you can work them into a simpler form.

Isolating Variables

The final crucial step is isolating the variable, a process where you aim to get the variable by itself on one side of the equation. This allows you to find its value.
Consider our simplified equation \(0.29x = 130.5\). To isolate \(x\), you need to eliminate the coefficient attached to it, which is \(0.29\).

The way to do this is by dividing both sides of the equation by \(0.29\).

• This operation gives you: \(x = \frac{130.5}{0.29}\).
• Once you perform the division, you get the value of \(x\), which is approximately \450\.

By isolating the variable, you transform an equation into its simplest form where the solution becomes evident. Mastering this step is essential for solving algebraic equations efficiently and with confidence.