Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{l}\frac{1}{x}+\frac{3}{y}=2 \\\ \frac{4}{x}-\frac{1}{y}=-5\end{array}\right.\)

Step-by-Step Solution

Answer

The exact solutions would require substituting \(y = \frac{2x}{3( x - 1)}\) into the original equations and solving for \(x\), which should then be substituted back to find \(y\). These should be verified graphically as it is part of the original question.

Step 1: Simplify the Equations to Similar Terms

Firstly, simplify the equations by finding a common denominator for each term. For the first equation, the common denominator is \(xy\), yielding \(y + 3x = 2xy\).The second equation simplifies to \(4y - x = -5xy\). These two equations become:\[ y + 3x - 2xy = 0 \]\[ 4y - x + 5xy = 0 \]

Step 2: Convert to a Linear Format

Next, rearrange the equations to look like quadratic equations. In this case, the two equations would be rearranged as follows:\[ 2xy - 3x - y = 0 \]\[ 5xy + x - 4y = 0 \]

Step 3: Elimination

For the elimination method, subtract the second equation from the first to eliminate \(x\).Subtracting the two equations yields:\[ (2xy - 5xy) - (3x - x) - (y - 4y) = 0\]Simplify to get:\[ -3xy - 2x + 3y = 0 \]

Step 4: Solve for y

Arranging the equation to solve for \(y\) yields:\[ y = \frac{2x}{3( x - 1)} \]Now substitute \(y\) into the first equation and solve for \(x\). This would be left as an exercise to perform the substitution and solving.

Step 5: Solve for x

Substitute the value of y obtained into either of the original equations and solve for \(x\). This value should be verified graphically.

Key Concepts

Elimination Method

The elimination method is a strategic process used to solve systems of equations by removing one variable, allowing for easier solution of the remaining equation. This method is especially helpful when dealing with linear equations.

Here's a basic rundown of the elimination method:

• Align the systems of equations vertically, ensuring that similar terms are aligned.
• If necessary, multiply one or both equations by a constant to ensure that when added or subtracted, one of the variables cancels out.
• Add or subtract the equations to eliminate one variable completely.
• Solve the resulting equation for the remaining variable.
• Substitute the value found back into one of the original equations to find the value of the eliminated variable.
• Check your solution by substituting the values into both original equations to ensure consistency.

Using the elimination method often provides a quick and clear path to the solution, particularly in algebra where finding common denominators might simplify the process further.

Linear Equations

Linear equations are fundamental in mathematics, representing relationships with constant rates of change. They manifest in the form of lines when graphed and possess the form:\[ ax + by = c \]where \(a\), \(b\), and \(c\) are constants.

They are characterized by:

• Constant coefficients, indicating no exponents other than 1 on any of the variables.
• Graphs that are straight lines.
• Intersections that, when plotted, may refer to solutions of systems of equations.

In the context of solving problems, systems of linear equations can often be addressed using either substitution, elimination, or graphically observed through plotting on a coordinate plane, using techniques such as the elimination method to simplify and solve.

Graphing Utility

A graphing utility is a digital tool or software application used to plot graphs and perform various calculations related to mathematical functions and equations. They are particularly useful for visually verifying the solutions of equations or inequalities.

Graphing utilities help by:

• Allowing for quick visualization of functions, making it easier to understand the behavior of mathematical equations.
• Offering features to find intersection points visually, which correspond to the solution of a system of equations.
• Facilitating the checking of algebraic work by providing a visual check against calculated solutions.

By inputting equations into a graphing utility, students can confirm analytical solutions with the plotted graph, enhancing understanding and confirming the accuracy of their answers.

Common Denominators

Finding common denominators is a crucial mathematical technique used to manage fractions within equations, making it easier to combine them or solve a system of equations. For our original problem, common denominators were key in rewiring the fractional equations into a solvable form.

Why common denominators matter:

• They allow for addition and subtraction of fractions to proceed smoothly, transforming them into simpler, unified algebraic expressions.
• Helps convert equations into a standard linear form, facilitating the use of methods like elimination or substitution.
• Aid in simplifying complex fractional equations to manageable linear styles, as seen when reducing to expressions with common terms.

In the process of solving a system by elimination, finding common denominators at the outset simplifies the algebraic manipulation, leading to more straightforward problem solving.