Solving the Algebraic Equation to Find the Value of x

The problem asks us to find the value of \(x\) that satisfies the given equation:

\[5\left( {1 - \frac{x}{5}} \right) - (5 - x) - \frac{1}{{200}}{\rm{of (20 - x) = 0}}{\rm{.08}}\]

To solve for \(x\), we need to simplify the equation and isolate the term containing \(x\).

Step-by-Step Solution Process

Let's break down the equation and simplify each part.

Step 1: Simplify the terms within the equation.

• Simplify the first term: \(5\left( {1 - \frac{x}{5}} \right)\) \[5 \times 1 - 5 \times \frac{x}{5} = 5 - x\]
• Simplify the second term: \(-(5 - x)\) \[-1 \times 5 - 1 \times (-x) = -5 + x\]
• Simplify the third term: \(-\frac{1}{{200}}{\rm{of (20 - x)}}\). Remember 'of' means multiplication. \[-\frac{1}{{200}} \times (20 - x) = -\left(\frac{1}{200} \times 20 - \frac{1}{200} \times x\right) = -\left(\frac{20}{200} - \frac{x}{200}\right)\] \[= -\left(\frac{1}{10} - \frac{x}{200}\right) = -\frac{1}{10} + \frac{x}{200}\] We can also write this in decimal form: \(-\left(0.1 - 0.005x\right) = -0.1 + 0.005x\).
• The right side of the equation is \(0.08\).

Step 2: Substitute the simplified terms back into the original equation.

The equation becomes:

\[(5 - x) + (-5 + x) + \left(-0.1 + 0.005x\right) = 0.08\]

Step 3: Combine like terms on the left side of the equation.

Combine the terms with \(x\):

\(-x + x + 0.005x = ( -1 + 1 + 0.005)x = 0.005x\)

Combine the constant terms:

\(5 - 5 - 0.1 = 0 - 0.1 = -0.1\)

So, the simplified equation is:

\[0.005x - 0.1 = 0.08\]

Step 4: Isolate the term with \(x\).

Add \(0.1\) to both sides of the equation:

\[0.005x - 0.1 + 0.1 = 0.08 + 0.1\]

\[0.005x = 0.18\]

Step 5: Solve for \(x\).

Divide both sides by \(0.005\):

\[x = \frac{0.18}{0.005}\]

To perform the division, we can multiply the numerator and the denominator by 1000 to remove the decimals:

\[x = \frac{0.18 \times 1000}{0.005 \times 1000} = \frac{180}{5}\]

Now, divide 180 by 5:

\[x = 36\]

The value of \(x\) that satisfies the given equation is 36.

Final Answer Check

Let's quickly check if \(x=36\) works in the original equation. This step is optional but helpful.

\(5\left( {1 - \frac{36}{5}} \right) - (5 - 36) - \frac{1}{{200}}{\rm{(20 - 36)}}\)

\(= 5\left( {1 - 7.2} \right) - (-31) - \frac{1}{{200}}{(-16)}\)

\(= 5\left( {-6.2} \right) + 31 - (-\frac{16}{200})\)

\(= -31 + 31 - (-\frac{2}{25})\)

\(= 0 + \frac{2}{25}\)

\(= \frac{8}{100} = 0.08\)

The left side equals \(0.08\), which matches the right side of the original equation. So, the value \(x=36\) is correct.

Revision Table: Key Steps in Solving Linear Equations

Understanding the general approach to solving linear equations is crucial.

Additional Information: Understanding 'of' in Mathematical Expressions

In mathematics, the word 'of' often indicates multiplication, especially when dealing with fractions or percentages. For example, 'half of 10' means \(\frac{1}{2} \times 10\), and '10% of 50' means \(\frac{10}{100} \times 50\). In the given equation, \(\frac{1}{200}{\rm{of (20 - x)}}\) means \(\frac{1}{200} \times (20 - x)\).

Understanding these basic conventions helps in correctly translating word problems and expressions into mathematical equations that can be solved.