Problem 15 - Elementary Algebra

Question

Solve each of the equations. $$0.07 x+0.08(x+600)=78$$

Step-by-Step Solution

Answer

x = 200.

Step 1: Simplifying the Equation

Let's start by distributing the 0.08 across the terms inside the parentheses. This gives us the equation: \[ 0.07x + 0.08x + 0.08 \times 600 = 78. \] Simplifying the terms results in:\[ 0.07x + 0.08x + 48 = 78. \]

Step 2: Combining Like Terms

Next, combine the like terms involving \(x\), which are the terms with \(0.07x\) and \(0.08x\).\[ (0.07 + 0.08)x + 48 = 78. \]This simplifies to:\[ 0.15x + 48 = 78. \]

Step 3: Isolating the Variable Term

To isolate the \(0.15x\) term, subtract 48 from both sides of the equation:\[ 0.15x + 48 - 48 = 78 - 48. \]This simplifies to:\[ 0.15x = 30. \]

Step 4: Solving for x

Now, solve for \(x\) by dividing both sides of the equation by 0.15:\[ x = \frac{30}{0.15}. \]Perform the division:\[ x = 200. \]

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions and solve equations more efficiently. It involves multiplying a single term by two or more terms inside a set of parentheses. Using the distributive property allows us to remove the parentheses by distributing, or multiplying, the outside term with each term inside. For example, in the given equation, we used the distributive property as follows:

• 0.08 is distributed across the terms in the parentheses, \( (x + 600) \), giving \( 0.08 \times x + 0.08 \times 600 \).
• Simplifying this distribution, we end up with \( 0.08x + 48 \).

By applying the distributive property, we're able to simplify the equation to a form where like terms can be easily combined. This step is essential and lays the groundwork for further simplification and for solving linear equations efficiently.

Combining Like Terms

Combining like terms is the process of simplifying expressions by merging terms that have the same variable raised to the same power. In the context of linear equations, it becomes crucial to streamline the equation to make it solvable. In our case, we combined like terms as follows:

• Identify terms that contain the variable \( x \), such as \( 0.07x \) and \( 0.08x \).
• Add these coefficients together, resulting in \( (0.07 + 0.08)x \).
• This simplifies to \( 0.15x \).

By combining like terms, we consolidate multiple terms into a simpler expression, making it much easier to proceed to the next steps in solving the equation. This technique is crucial in breaking down complex expressions into manageable forms.

Solving Linear Equations

Solving linear equations involves finding the value of the unknown variable that satisfies the equation. Once the equation is simplified through distribution and combining like terms, the next step is manipulating the equation to isolate the variable. Here's how it works in the example:

• Start with the simplified equation \( 0.15x + 48 = 78 \).
• To isolate the term involving \( x \), subtract 48 from both sides: \( 0.15x = 30 \).
• Finally, solve for \( x \) by dividing both sides by 0.15 resulting in \( x = \frac{30}{0.15} \).
• Perform the division to find \( x = 200 \).

This step is about systematically organizing the equation to make the variable stand alone on one side. Once that's done, you perform the arithmetic operation required to solve for the unknown. This method is repeatable and reliable for any linear equation, which means it is a mainstay in algebra.