Problem 35 - Precalculus: Mathematical for Calculus

Question

Find all solutions of the system of equations. $$\left\\{\begin{array}{c}\frac{2}{x}-\frac{3}{y}=1 \\\\-\frac{4}{x}+\frac{7}{y}=1\end{array}\right.$$

Step-by-Step Solution

Answer

The solution is \((x, y) = \left( \frac{1}{5}, \frac{1}{3} \right)\).

Step 1: Eliminate Variable x

First, we want to eliminate one of the variables by using linear combinations of the given equations. Multiply the first equation by 2.5 and the second equation by 1.25 to have equal coefficients of \( \frac{1}{x} \). This results in: \[ 2.5 \left( \frac{2}{x} - \frac{3}{y} = 1 \right) \rightarrow \frac{5}{x} - \frac{7.5}{y} = 2.5 \] \[ 1.25 \left( -\frac{4}{x} + \frac{7}{y} = 1 \right) \rightarrow -\frac{5}{x} + \frac{8.75}{y} = 1.25 \]. Add these two resulting equations to eliminate \( \frac{1}{x} \).

Step 2: Simplify the Resulting Equation

When we add the new equations from Step 1, \( \frac{5}{x} \) cancels out, and we get: \[ \left( \frac{8.75}{y} - \frac{7.5}{y} = 1.25 + 2.5 \right) \rightarrow \frac{1.25}{y} = 3.75 \]. Solve for \( \frac{1}{y} \): \[ \frac{1}{y} = \frac{3.75}{1.25} = 3 \]. Thus, \( y = \frac{1}{3} \).

Step 3: Substitute y to Find x

Now substitute \( y = \frac{1}{3} \) back into one of the original equations to solve for \( x \). Using the first equation: \[ \frac{2}{x} - \frac{3}{\frac{1}{3}} = 1 \]. This simplifies to: \[ \frac{2}{x} - 9 = 1 \Rightarrow \frac{2}{x} = 10 \]. Solve for \( x \): \[ x = \frac{1}{5} \].

Step 4: Verify the Solutions

Substitute \( x = \frac{1}{5} \) and \( y = \frac{1}{3} \) back into the original equations to ensure they satisfy both. For the first equation: \[ \frac{2}{\frac{1}{5}} - \frac{3}{\frac{1}{3}} = 10 - 9 = 1 \]. For the second equation: \[ -\frac{4}{\frac{1}{5}} + \frac{7}{\frac{1}{3}} = -20 + 21 = 1 \]. Both equations hold true, thus \((x, y) = \left( \frac{1}{5}, \frac{1}{3} \right)\) is the solution to the system.

Key Concepts

Linear Equations

A linear equation is an equation that creates a straight line when graphed. It represents a relationship between two variables. In a system of equations, we're dealing with two or more linear equations. Each of these equations expresses a line in a specific form. In the given problem, we have two equations that involve fractions:

• \( \frac{2}{x} - \frac{3}{y} = 1 \)
• \( -\frac{4}{x} + \frac{7}{y} = 1 \)

These equations can be complicated by the presence of fractional coefficients. The goal is to find values of \( x \) and \( y \) that simultaneously satisfy both equations. Understanding the structure of these equations helps in applying strategies like variable elimination to solve the system.
When approaching linear equations, remember:

• They can be manipulated by multiplying both sides by constants to simplify or align coefficients.
• This alignment is key to canceling out variables and reducing complexity.

Variable Elimination

Variable elimination is a technique used to solve a system of equations by removing one of the variables. This simplifies the process to find the remaining variable.In the given problem, we used variable elimination by aligning the coefficients of \( \frac{1}{x} \) in both equations:

• Multiply the first equation by 2.5: \( \frac{5}{x} - \frac{7.5}{y} = 2.5 \)
• Multiply the second equation by 1.25: \( -\frac{5}{x} + \frac{8.75}{y} = 1.25 \)
• Adding these equations cancels out \( \frac{5}{x} \), simplifying the system.

This step leads us to an equation in one variable, y, which can be solved directly. Variable elimination often involves multiplying one or both equations to align a particular coefficient, allowing for straightforward addition or subtraction to eliminate a variable. This technique is powerful because it reduces a two-variable system to a single-variable equation.

Solution Verification

Verifying solutions ensures the accuracy of the answers obtained from solving equations. It's the process of checking whether the solutions actually satisfy the original equations. In the given problem, once we solve for \( x = \frac{1}{5} \) and \( y = \frac{1}{3} \), we substitute these values back into both original equations.
For the first equation:

• \( \frac{2}{\frac{1}{5}} - \frac{3}{\frac{1}{3}} = 10 - 9 = 1 \)

For the second equation:

• \( -\frac{4}{\frac{1}{5}} + \frac{7}{\frac{1}{3}} = -20 + 21 = 1 \)

Both checks are true, confirming that our solution \((x, y) = \left( \frac{1}{5}, \frac{1}{3} \right)\) is correct. Solution verification is crucial not just for peace of mind but to ensure that the mathematical manipulations done during solving, like variable elimination, didn't introduce errors.