Question

follow the step - by - step process to solve the linear equation
a) what is the smallest power of 10 that can be used to eliminate all decimals after multiplying both sides of the original equation by that power of 10?
b) write the new linear equation obtained after multiplying both sides of the original equation by the smallest power of 10 determined in part (a).
c) solve the equation.
\\(0.002(6 - k)+0.01(k - 2)=1\\)
a) the smallest power of 10 is \\(\square\\)
(type an integer or a decimal)
b) what is the new equation?
\\(\bigcirc\\) a. \\(2(6 - k)+10(k - 2)=1\\)
\\(\bigcirc\\) b. \\(2(6 - k)+10(k - 2)=1000\\)
\\(\bigcirc\\) c. \\(2(6000 - k)+10(k - 2000)=1000\\)
\\(\bigcirc\\) d. \\(2(6 - k)+1(k - 2)=1\\)
c) what is the solution? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. \\(k = \square\\) (type an integer or a simplified fraction)
\\(\bigcirc\\) b. the solution is all real numbers
\\(\bigcirc\\) there is no solution

Explanation:

Part (a)

Step1: Identify decimal places

The decimals are \(0.002\) (3 decimal places) and \(0.01\) (2 decimal places). The maximum number of decimal places is 3.

Step2: Determine power of 10

To eliminate 3 decimal places, we use \(10^3 = 1000\).

Step1: Multiply by 1000

Multiply each term in \(0.002(6 - k)+0.01(k - 2)=1\) by \(1000\).

Step2: Simplify each term

• \(1000\times0.002(6 - k)=2(6 - k)\)
• \(1000\times0.01(k - 2)=10(k - 2)\)
• \(1000\times1 = 1000\)

So the new equation is \(2(6 - k)+10(k - 2)=1000\), which matches option B.

Step1: Expand the equation

Expand \(2(6 - k)+10(k - 2)=1000\):
\(12-2k + 10k-20 = 1000\)

Step2: Combine like terms

Combine \( -2k+10k = 8k\) and \(12 - 20=-8\), so \(8k-8 = 1000\)

Step3: Solve for \(k\)

Add 8 to both sides: \(8k=1000 + 8=1008\)
Divide by 8: \(k=\frac{1008}{8}=126\)

Answer:

\(1000\) (or \(10^3\))

Part (b)