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.
\\(4.18x + 2.34 = 3.4x\\)

Explanation:

Part (a)

Step1: Analyze decimal places

The decimals in the equation are \(4.18\) (two decimal places), \(2.34\) (two decimal places), and \(3.4\) (one decimal place). To eliminate all decimals, we need to consider the maximum number of decimal places, which is 2.

Step2: Determine power of 10

A power of 10 with exponent equal to the number of decimal places (2) will eliminate the decimals. So \(10^2 = 100\).

Step1: Multiply each term by 100

Multiply \(4.18x\), \(2.34\), and \(3.4x\) by \(100\). Using the property \(a\times10^n\) where \(n = 2\) (for two decimal places), we get \(100\times4.18x+100\times2.34 = 100\times3.4x\).

Step2: Calculate each term

\(100\times4.18x = 418x\), \(100\times2.34 = 234\), and \(100\times3.4x = 340x\). So the new equation is \(418x + 234 = 340x\).

Step1: Subtract \(418x\) from both sides

To get all \(x\) terms on one side, subtract \(418x\) from both sides: \(418x + 234 - 418x = 340x - 418x\).

Step2: Simplify both sides

Left side: \(234\), Right side: \(-78x\). So we have \(234=-78x\).

Step3: Solve for \(x\)

Divide both sides by \(-78\): \(x=\frac{234}{-78}\).

Step4: Simplify the fraction

\(\frac{234}{-78}=-3\).

Answer:

\(100\)

Part (b)