Problem 16 - Elementary Algebra

Question

Solve each of the equations. $$0.06 x+0.09(x+200)=63$$

Step-by-Step Solution

Answer

The solution is \(x = 300\).

Step 1: Distribute the 0.09

Apply the distributive property to the term \(0.09(x+200)\). This means multiplying both \(x\) and \(200\) by \(0.09\). This results in \(0.09x + 18\).

Step 2: Rewrite the Equation

Substitute the expression from Step 1 back into the original equation, resulting in: \(0.06x + 0.09x + 18 = 63\)

Step 3: Combine Like Terms

Combine the \(x\) terms. \(0.06x + 0.09x = 0.15x\)This changes the equation to: \(0.15x + 18 = 63\)

Step 4: Isolate the Variable Term

Subtract 18 from both sides to isolate the \(0.15x\) term. \(0.15x + 18 - 18 = 63 - 18\)Thus, the equation simplifies to: \(0.15x = 45\).

Step 5: Solve for x

Divide both sides by 0.15 to solve for \(x\). \(x = \frac{45}{0.15}\)Thus, \(x = 300\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in elementary algebra. It's like when you share snacks with your friends, you need to give some to everyone, ensuring each friend gets their fair share! Mathematically, it involves multiplying a single term by each term inside a set of parentheses. In the exercise, we see \(0.09(x+200)\). Using the distributive property, we multiply \(0.09\) by both \(x\) and \(200\), resulting in \(0.09x + 18\). This step helps to break down expressions, making them simpler and easier to work with.
A way to remember the distributive property is the "rainbow" method – think of the multiplier (here, \(0.09\)) as the rainbow that touches each item in the parentheses. This property not only simplifies calculations but also prepares the equation for further simplification and solving!

Combining Like Terms

Combining like terms is the process of merging terms in an equation with the same variable or power. Think of it like organizing your workspace. You wouldn't mix up your pens with your pencils! Similarly, in the equation, we have like terms \(0.06x\) and \(0.09x\). These terms both have the variable \(x\), so we can add them together to get \(0.15x\).
By combining like terms, we simplify the equation, reducing the number of terms and making it more manageable. This step not only organizes the equation but also plays a crucial role in moving towards finding the variable's value. Remember, combining like terms is all about neatness and keeping similar terms together!

Isolating Variables

Isolating variables is a key step in solving algebraic equations. It's akin to finding the main character in a story and focusing on them to reveal the plot. Here, we are trying to solve for \(x\), which means we need to isolate \(x\) on one side of the equation.
In the example, after combining like terms, we have \(0.15x + 18 = 63\). To isolate \(x\), we first subtract \(18\) from both sides. This cancels out the \(+18\) on the left and adjusts the number on the right, resulting in \(0.15x = 45\). The next step is dividing both sides by \(0.15\) to finally solve for \(x\).
This process shows that isolating the variable step by step reduces complexity and helps to find the unknown value. It's like simplifying the path on a treasure map to reach the final destination, where \(x\) reveals itself as \(300\) in this equation!