Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the system. $$ \left\\{\begin{array}{l} \frac{3}{x-1}+\frac{4}{y+2}=2 \\ \frac{6}{x-1}-\frac{7}{y+2}=-3 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = 5 \) and \( y = 0 \).

Step 1: Define Variables

Let us define the variables as given in the problem. Let \( a = x - 1 \) and \( b = y + 2 \). This means \( x = a + 1 \) and \( y = b - 2 \).

Step 2: Substitute and Simplify Equations

Substitute \( a \) and \( b \) into the equations: \[ \frac{3}{a} + \frac{4}{b} = 2 \] and \[ \frac{6}{a} - \frac{7}{b} = -3 \]. Now we have a system of equations in terms of \( a \) and \( b \).

Step 3: Solve for 'a' in Terms of 'b' from First Equation

From the equation \( \frac{3}{a} + \frac{4}{b} = 2 \), isolate \( \frac{3}{a} \), giving \( \frac{3}{a} = 2 - \frac{4}{b} \).

Step 4: Substitute 'a' into the Second Equation

Rewrite \( \frac{3}{a} = 2 - \frac{4}{b} \) as \( a = \frac{3}{2 - \frac{4}{b}} \) and substitute into the second equation. Calculate \( \frac{6}{a} = 2 \times \frac{3}{a} = 2 \times (2 - \frac{4}{b}) \).

Step 5: Solve for 'b'

Substitute and simplify the expression \( \frac{6}{a} - \frac{7}{b} = -3 \) using \( a = \frac{3}{2 - \frac{4}{b}} \): \( 2 \times (2 - \frac{4}{b}) - \frac{7}{b} = -3 \). Solve for \( b \).

Step 6: Solve for 'a' Using 'b'

After calculating \( b \), substitute it back into the expression \( a = \frac{3}{2 - \frac{4}{b}} \) to find the value of \( a \).

Step 7: Calculate 'x' and 'y'

Having values for \( a \) and \( b \), calculate \( x = a + 1 \) and \( y = b - 2 \).

Step 8: Present the Solution

The solution for the system of equations \( \begin{cases} \frac{3}{x-1} + \frac{4}{y+2} = 2 \ \frac{6}{x-1} - \frac{7}{y+2} = -3 \end{cases} \) is \( x \) and \( y \) from the calculations in steps above.

Key Concepts

Substitution Method

The substitution method is a popular technique for solving systems of equations. It involves expressing one variable in terms of the other using one equation, and then substituting this expression into another equation.
By doing this, we effectively reduce the number of variables, allowing us to solve the system more easily.

This method is especially useful when dealing with rational equations, as it helps simplify complex expressions.

• In our given problem, we start by defining new variables, \(a = x - 1\) and \(b = y + 2\).
  This redefinition is a form of substitution that helps simplify the rational expressions \(\frac{3}{x-1}\) and \(\frac{4}{y+2}\).
• After substituting \(a\) and \(b\) into the equations, we continue by expressing \(a\) in terms of \(b\) from one equation.
  This way, we can substitute \(a\) when working with the second equation.

This approach helps streamline the calculation process and systematically reach a solution by focusing on one variable at a time.

Algebraic Manipulation

Algebraic manipulation is the art of rearranging equations to isolate variables, simplify expressions, and solve systems of equations. It involves operations like addition, subtraction, multiplication, division, and applying properties of equalities to transform equations.
Through these manipulations, we can find solutions with greater ease.

Let's see how it applies in our example:

• The first step after substitution is to isolate terms. For \(\frac{3}{a} + \frac{4}{b} = 2\), we isolate \(\frac{3}{a}\) by subtracting \(\frac{4}{b}\), leading to \(\frac{3}{a} = 2 - \frac{4}{b}\).
• Algebraic manipulation continues as we focus on transforming \(a = \frac{3}{2 - \frac{4}{b}}\).
• Such manipulations allow us to substitute expressions consistently into other parts of the system, simplifying the process of finding solutions.
  In this setup, careful manipulation helps us express both \(a\) and \(b\) in solvable forms.

This technique complements the substitution method perfectly and can drastically simplify complex rational equations.

Rational Equations

Rational equations are equations involving ratios of polynomials, often requiring keen attention to detail when solving due to their complexity. The denominator in these equations can complicate solving them directly. However, carefully handling these rational equations helps avoid common algebraic pitfalls.

In the example provided, the expressions \(\frac{3}{x-1}\) and \(\frac{4}{y+2}\) denote rational parts of the system.

• The solution approach begins with substitution to define simpler variables (\(a\) and \(b\)), allowing us to clear these fractions for manageable expressions.
• By setting up equations like \(\frac{3}{a} + \frac{4}{b} = 2\), we simplify solving the rational system. We then utilize algebraic manipulation to isolate and solve for \(a\) and \(b\).

Addressing rational equations involves dealing carefully with denominators to prevent undefined values and ensuring all subsequent operations are valid.
This method ensures a detailed and systematic path to solutions, enhancing comprehension and skill in handling similar equations.