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Chapter 2: Problem 69

Solve each equation, and check your solution. $$ 0.006(x+2)=0.007 x+0.009 $$

Short Answer

Expert verified
x = 3

Step by step solution

01

Distribute the 0.006

First, distribute the 0.006 across the terms inside the parentheses: \[ 0.006(x + 2) = 0.006x + 0.012 \]So the equation becomes:\[ 0.006x + 0.012 = 0.007x + 0.009 \]
02

Isolate the variable terms on one side

Subtract 0.006x from both sides to start isolating x:\[ 0.012 = 0.001x + 0.009 \]
03

Isolate x

Subtract 0.009 from both sides:\[ 0.012 - 0.009 = 0.001x \]This simplifies to:\[ 0.003 = 0.001x \]
04

Solve for x

Divide both sides by 0.001:\[ x = \frac{0.003}{0.001} = 3 \]
05

Check the solution

Substitute x = 3 back into the original equation to check:\[ 0.006(3 + 2) = 0.007(3) + 0.009 \]This gives:\[ 0.006(5) = 0.021 + 0.009 \]\[ 0.03 = 0.03 \]Since both sides are equal, x = 3 is correct.

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Key Concepts

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

Distribution in Algebra
When solving linear equations, one useful technique is distribution.
Distribution means to multiply each term inside parentheses by a factor outside of them.
For instance, in the given problem, we have the term \(0.006(x+2)\).
To eliminate the parentheses, we multiply 0.006 by both \(x\) and 2:
\[ 0.006 \times x = 0.006x \]
\[0.006 \times 2 = 0.012 \].
This helps simplify the equation, making it easier to solve.
In general, always remember that:
\[ a(b + c) = ab + ac \]
This principle is used extensively in algebra to clear parentheses and simplify expressions.
Isolating the Variable
After distributing, the next key step is to isolate the variable.
The variable is the unknown value we are trying to solve for, often represented by \(x\).
To isolate the variable, we need to get it alone on one side of the equation.
Here鈥檚 a systematic way to do this:
  • First, move all variable terms to one side. In our problem, we subtract 0.006x from both sides to get: \[ 0.012 = 0.001x + 0.009 \]
  • Next, move the constant term to the other side. Subtract 0.009 from both sides:
    \[ 0.012 - 0.009 = 0.001x \]
  • Finally, divide by the coefficient of x to solve for it:
    \[ x = \frac{0.003}{0.001} = 3 \]

Remember, each step requires performing the same operation on both sides of the equation.
Checking Solutions
It鈥檚 good practice to check your solution to ensure it鈥檚 correct.
To do this, substitute the value you found back into the original equation.
For our example, substituting x = 3 gives:
\[ 0.006(3 + 2) = 0.007(3) + 0.009 \]
Simplify both sides:
\[ 0.006 \times 5 = 0.03 \]
\[ 0.021 + 0.009 = 0.03 \]
Since both sides of the equation are equal, the solution is confirmed.
This step is crucial as it validates your work and ensures no mistakes were made during the calculations.

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