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Chapter 2: Problem 56

Solve each equation. $$0.02(x-100)+0.06 x=62$$

Short Answer

Expert verified
x = 800

Step by step solution

01

Distribute the 0.02

First, distribute the 0.02 across the terms inside the parentheses:the equation becomes \[ 0.02x - 2 + 0.06x = 62 \]
02

Combine Like Terms

Next, combine the like terms (terms with x):\[ 0.02x + 0.06x - 2 = 62 \]These combine to \[ 0.08x - 2 = 62 \]
03

Add 2 to Both Sides

Add 2 to both sides of the equation to isolate the term with x:\[ 0.08x - 2 + 2 = 62 + 2 \]This simplifies to\[ 0.08x = 64 \]
04

Solve for x

Finally, divide both sides by 0.08 to solve for x:\[ \frac{0.08x}{0.08} = \frac{64}{0.08} \]This gives \[ x = 800 \]

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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 useful when you need to eliminate parentheses in an equation. It's all about spreading out the multiplication over addition or subtraction. For instance, consider the problem: 0.02(x-100) + 0.06x = 62.
We distribute 0.02 to both x and -100 inside the parentheses. This means: 0.02(x) + 0.02(-100).
After distributing, we get: 0.02x - 2. So, our equation turns into: 0.02x - 2 + 0.06x = 62.
Using the distributive property simplifies the equation, making it easier to solve step-by-step.
Combining Like Terms
After using the distributive property, our next goal is to make the equation simpler. This is where we combine like terms. Like terms are numbers or variables that have the same letters and exponents.
In our equation 0.02x - 2 + 0.06x = 62, notice how 0.02x and 0.06x are like terms because they both contain the variable x.
Combining these gives us 0.08x, so the equation becomes: 0.08x - 2 = 62.
Combining like terms reduces the number of terms in the equation, making it easier to handle.
Isolating the Variable
The last important step in solving a linear equation is isolating the variable, which means getting the variable alone on one side of the equation. In our simplified equation, 0.08x - 2 = 62, the variable is x.
We need to get x by itself.
First, add 2 to both sides: 0.08x - 2 + 2 = 62 + 2 simplifies to 0.08x = 64.
Next, we divide both sides by 0.08: 0.08x/0.08 = 64/0.08. This gives us x = 800.
Isolating the variable allows us to find its value easily.

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