Solving for \(x^3 + y^3\) using Given Equations

We are given two equations involving variables \(x\) and \(y\):

• Equation 1: \(x + y = 2\)
• Equation 2: \(\frac{1}{x} + \frac{1}{y} = \frac{18}{5}\)

Our goal is to find the value of the expression \(x^3 + y^3\).

Step 1: Simplify the Second Equation to Find \(xy\)

The second equation involves reciprocals. We can combine the terms on the left-hand side by finding a common denominator, which is \(xy\):

\(\frac{1}{x} + \frac{1}{y} = \frac{y}{xy} + \frac{x}{xy} = \frac{y+x}{xy}\)

So, Equation 2 becomes:

\(\frac{x+y}{xy} = \frac{18}{5}\)

Now, we can use Equation 1 (\(x+y=2\)) to substitute the value of \(x+y\) into this simplified equation:

\(\frac{2}{xy} = \frac{18}{5}\)

To solve for \(xy\), we can cross-multiply:

\(2 \times 5 = 18 \times xy\)

\(10 = 18xy\)

Divide both sides by 18:

\(xy = \frac{10}{18}\)

Simplify the fraction:

\(xy = \frac{5}{9}\)

So, we have found the value of \(xy\).

Step 2: Calculate \(x^3 + y^3\) using an Algebraic Identity

We need to find the value of \(x^3 + y^3\). There is a useful algebraic identity for the sum of cubes:

\(x^3 + y^3 = (x+y)(x^2 - xy + y^2)\)

Alternatively, we can use another form which is often more convenient when \(x+y\) and \(xy\) are known:

\(x^3 + y^3 = (x+y)^3 - 3xy(x+y)\)

We know the values of \(x+y\) and \(xy\) from our previous steps:

• \(x+y = 2\)
• \(xy = \frac{5}{9}\)

Substitute these values into the identity:

\(x^3 + y^3 = (2)^3 - 3 \left(\frac{5}{9}\right) (2)\)

Calculate the terms:

\((2)^3 = 8\)

\(3 \left(\frac{5}{9}\right) (2) = 3 \times \frac{10}{9} = \frac{30}{9}\)

Simplify the fraction \(\frac{30}{9}\) by dividing the numerator and denominator by their greatest common divisor, which is 3:

\(\frac{30}{9} = \frac{30 \div 3}{9 \div 3} = \frac{10}{3}\)

Now substitute these values back into the expression for \(x^3 + y^3\):

\(x^3 + y^3 = 8 - \frac{10}{3}\)

To subtract these values, find a common denominator, which is 3. Convert 8 to a fraction with denominator 3:

\(8 = \frac{8 \times 3}{1 \times 3} = \frac{24}{3}\)

Now perform the subtraction:

\(x^3 + y^3 = \frac{24}{3} - \frac{10}{3} = \frac{24 - 10}{3} = \frac{14}{3}\)

Step 3: Convert the Result to a Mixed Number

The value of \(x^3 + y^3\) is \(\frac{14}{3}\). We can convert this improper fraction to a mixed number to match the format of the options.

Divide 14 by 3:

\(14 \div 3 = 4\) with a remainder of \(14 - (3 \times 4) = 14 - 12 = 2\).

So, the mixed number is \(4 \frac{2}{3}\).

Thus, the value of \(x^3 + y^3\) is \(4 \frac{2}{3}\).

Revision Table: Key Algebraic Concepts

Additional Information: Solving Algebraic Problems

Algebraic problems like this one often require recognizing patterns and using known identities or formulas. Here are some tips:

• Simplify Equations: Before substituting or solving, try to simplify the given equations. Combining fractions, expanding expressions, or factoring can make the problem easier.
• Identify Needed Values: Determine what specific values (like \(x+y\), \(xy\), \(x^2+y^2\), etc.) would be useful for finding the target expression.
• Use Algebraic Identities: Memorizing key identities (like sum/difference of squares, cubes, perfect squares) is crucial as they provide shortcuts.
• Systematic Steps: Break down the problem into smaller, manageable steps. For example, first find \(xy\), then use it to find \(x^3+y^3\).
• Check Your Work: After finding a solution, quickly review your calculations to catch any arithmetic errors.

In this problem, recognizing that \(\frac{1}{x} + \frac{1}{y}\) simplifies to \(\frac{x+y}{xy}\) and knowing the identity for \(x^3 + y^3\) in terms of \(x+y\) and \(xy\) were key to finding the solution efficiently.