Problem 86 - Algebra for College Students

Question

Solve each system. \(\left(\begin{array}{l}\frac{3}{x}+\frac{2}{y}=2 \\\ \frac{2}{x}-\frac{3}{y}=\frac{1}{4}\end{array}\right)\)

Step-by-Step Solution

Answer

The solution is \( x = \frac{4}{3} \) and \( y = 4 \).

Step 1: Identify the Variables

Let's denote \( \frac{1}{x} \) as \( a \) and \( \frac{1}{y} \) as \( b \). Thus, the given system of equations becomes: \( 3a + 2b = 2 \) and \( 2a - 3b = \frac{1}{4} \).

Step 2: Express One Variable in Terms of the Other

From the first equation, express \( a \) in terms of \( b \): \( 3a = 2 - 2b \). Therefore, \( a = \frac{2 - 2b}{3} \).

Step 3: Substitute into the Second Equation

Substitute \( a = \frac{2 - 2b}{3} \) into the second equation \( 2a - 3b = \frac{1}{4} \), giving: \( 2 \left( \frac{2 - 2b}{3} \right) - 3b = \frac{1}{4} \).

Step 4: Clear Fractions and Simplify

Multiply the entire equation by 12 to clear denominators: \( 8(2 - 2b) - 36b = 3 \). Simplify to \( 16 - 16b - 36b = 3 \), resulting in \(-52b = -13 \).

Step 5: Solve for b

Divide both sides by -52 to get \( b = \frac{1}{4} \).

Step 6: Substitute b Back to Find a

Substitute \( b = \frac{1}{4} \) back into \( a = \frac{2 - 2b}{3} \): \( a = \frac{2 - 2 \times \frac{1}{4}}{3} \). This simplifies to \( a = \frac{3}{4} \).

Step 7: Find x and y

Since \( a = \frac{1}{x} \) and \( b = \frac{1}{y} \), we find \( x = \frac{4}{3} \) and \( y = 4 \).

Key Concepts

Variables Substitution

In linear equations, sometimes it's helpful to transform variables to make the equations simpler to work with. By introducing new variables, the problem often becomes more manageable. In this exercise, we introduce variable substitution by letting \( \frac{1}{x} = a \) and \( \frac{1}{y} = b \). This substitution changes the system of equations from fractions into a more familiar linear form.

This transformation makes it easier to solve because you can now work with whole numbers instead of fractions. This is particularly useful when dealing with equations involving complex fractions, like \( \frac{3}{x} + \frac{2}{y} = 2 \).

Once substitution has been successfully applied, the equations become:

• \( 3a + 2b = 2 \)
• \( 2a - 3b = \frac{1}{4} \)

Now, the equations are simpler, facilitating easier manipulation in subsequent steps.

Clearing Fractions

Fractions can make algebra seem tricky, but there are methods to eliminate them, simplifying the process. The key is clearing fractions by multiplying the entire equation by a suitable number. Here, the goal is to remove the denominators completely.

In this scenario, we have the equation after substitution as \( 2 \left( \frac{2 - 2b}{3} \right) - 3b = \frac{1}{4} \). To eliminate fractions, multiply the whole equation by 12, the least common multiple of all denominators involved (3 and 4).

This step gives:

• \( 8(2 - 2b) - 36b = 3 \)

This procedure simplifies the equation by removing fractions, allowing for straightforward algebraic manipulation in subsequent steps.

Solving Equations Step-by-Step

Breaking down problems into simpler, more manageable steps can help find solutions efficiently. After clearing fractions, the next task is isolating and solving variables.

Following the transformation and clearing fractions, the equation \( 8(2 - 2b) - 36b = 3 \) results in a much simpler equation.

As you simplify:

• First, expand: \( 16 - 16b - 36b = 3 \).
• Then, combine like terms: \(-52b = -13 \).
• Finally, divide by -52 to isolate \( b \): \( b = \frac{1}{4} \).

Each of these steps is about working systematically. By isolating each term, and then each variable, you keep track of your solutions and ensure correctness before progressing.

Algebraic Simplification

Simplification is a key part of solving equations. It means making the math less cluttered and easier to observe, whenever possible.

After establishing \( b = \frac{1}{4} \), substitute back to find \( a \) with the equation \( a = \frac{2 - 2b}{3} \). Plugging \( b = \frac{1}{4} \) gives:

• \( a = \frac{2 - 2 \times \frac{1}{4}}{3} \)
• \( a = \frac{3}{4} \)

Simplification allows you to progress from an expression to a simple format, making it easier to find solutions. Finally, since \( a = \frac{1}{x} \) and \( b = \frac{1}{y} \), it gives you \( x = \frac{4}{3} \) and \( y = 4 \). With careful checking, this simplification confirms the algebraic relationships found between variables.