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Chapter 2: Problem 55

Solve each equation. $$0.03(x+200)+0.05 x=86$$

Short Answer

Expert verified
x = 1000

Step by step solution

01

Distribute the 0.03 inside the parentheses

First, distribute the 0.03 to both terms inside the parentheses: \[ 0.03(x + 200) + 0.05x = 86 \] This becomes: \[ 0.03x + 6 + 0.05x = 86 \]
02

Combine like terms

Combine the terms with \(x\): \[ 0.03x + 0.05x + 6 = 86 \] This simplifies to: \[ 0.08x + 6 = 86 \]
03

Subtract 6 from both sides

To isolate the term with \(x\), subtract 6 from both sides of the equation: \[ 0.08x + 6 - 6 = 86 - 6 \] This results in: \[ 0.08x = 80 \]
04

Divide both sides by 0.08

Finally, divide both sides of the equation by 0.08 to solve for \(x\): \[ x = \frac{80}{0.08} \] This gives: \[ x = 1000 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a useful tool in algebra. It allows you to simplify expressions. When you have something like \(a(b + c)\), the property lets you distribute the \(a\) to both \(b\) and \(c\). This gives you \(ab + ac\). Let's see it in action in our exercise.

In the equation \[0.03(x + 200) + 0.05x = 86\], we use the distributive property to simplify \[0.03(x + 200)\].
We get: \[0.03x + 0.03 \times 200\]. When simplified, this becomes \[0.03x + 6\].

Now our equation looks like \[0.03x + 6 + 0.05x = 86 \] . By using the distributive property, we made our equation simpler!
Combining Like Terms
Once we have distributed, our next step is to combine like terms. Like terms are terms that have the same variable raised to the same power.

In our equation, after distribution, we have \[0.03x + 6 + 0.05x = 86\].
The terms \(0.03x\) and \(0.05x\) are like terms because they both have the \(x\) variable.

We combine them to get \[0.03x + 0.05x = 0.08x\].
This changes our equation to \[0.08x + 6 = 86\]. By combining like terms, we simplify it further.
Isolation of Variable
Now that our equation is simpler, it's time to 'isolate the variable' which means to get \(x\) by itself on one side of the equation.

Starting from \[0.08x + 6 = 86\], we'll subtract 6 from both sides:
\[0.08x + 6 - 6 = 86 - 6\] simplifies to \[0.08x = 80\].

Now we need to solve for \(x\) by dividing both sides by 0.08: \[x = \frac{80}{0.08} = 1000\].

We found that \(x = 1000\). By isolating the variable, we solved the equation!

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