Problem 34 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{l}\frac{2}{3} x+\frac{1}{6} y=\frac{2}{3} \\ 4 x+y=4\end{array}\right.\)

Step-by-Step Solution

Answer

The system of equations has infinite solutions, expressed generally as \(y = 4 - 4x\), for all real \(x\).

Step 1: Equalize Coefficients of one Variable

To make the coefficients of \(y\) match in both equations, we multiply the first equation by \(6\) and the second one by \(1\). This produces the following system: \[\left\{\begin{array}{l} 4x+y=4 \ 4x + y = 4\end{array}\right.\]

Step 2: Deduction

Subtracting the second equation from the first one effectively removes \(y\), which results in \(0=0\). This means these two equations are identical, and the system of equations has infinite solutions.

Step 3: Find the general solution

Since the two equations are identical, we can express \(y\) in terms of \(x\), which is done by rearranging the second equation: \(y = 4 - 4x\). Therefore, every pair \((x, 4-4x)\) for \(x \in \mathbb{R}\) is a solution to this system.

Key Concepts

Method of Elimination

The method of elimination is a technique used to solve systems of linear equations. This method involves aligning equations so that adding or subtracting them cancels out one of the variables. The aim is to reduce the system to a single equation with one variable, which can be solved easily.

In the case of the exercise, we attempted to align the coefficients of y to proceed with elimination. However, this led to the discovery that both equations were actually identical once we multiplied the first equation by 6. Consequently, subtracting one from the other resulted in an identity, indicating that the equations do not compete, but coincide, leading to infinite solutions.

System of Equations

A system of equations is a set of two or more equations with the same variables. The solution to a system is the set of variable values that satisfies all equations simultaneously. There are various types of solutions a system may have:

• A unique solution (the lines intersect at one point)
• No solution (the lines are parallel)
• Infinite solutions (the lines are coincident)

Our textbook exercise revealed a system with infinite solutions, as both equations represent the same line in a two-dimensional space. Graphically, this means they overlap perfectly along all points on the line.

Graphing Utility

A graphing utility is a tool—often a software application or graphing calculator—that allows users to visually represent equations as graphs. It serves as a valuable aid in understanding the nature of solutions to a system of equations.

In practice, after solving the equations algebraically, we can use a graphing utility to confirm our solution by plotting the lines represented by the equations. If the lines coincide, as they would in our example with infinite solutions, the visual confirmation aligns with our algebraic findings.

Algebraic Manipulation

Algebraic manipulation involves the rearrangement and simplification of equations using algebraic rules. The goal is to transform complex equations into simpler forms to identify solutions or other characteristics of the system.

In the original exercise, we engaged in algebraic manipulation when we multiplied the first equation by 6 in an attempt to make the y coefficients match for elimination. Proper manipulation is crucial, as errors in this step can lead to incorrect conclusions about the nature of the system's solutions.

Infinite Solutions

A system of equations presents infinite solutions when all sets of variable values satisfy both equations. This usually occurs when the equations are different expressions of the same relationship, essentially describing the same line.

Infinite solutions indicate that you can plug any number into one of the equations and use it to find a corresponding value in the other equation that will hold true. As seen in the exercise, by expressing y in terms of x, we acknowledge that every pair \[\[\begin{align*}(x, 4-4x)\end{align*}\]\]for \[\[\begin{align*}x ∈ ℜ\end{align*}\]\]is a valid solution, which means there's an infinite number of points along the line where the two equations intersect.