Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{aligned} \frac{1}{4} x+\frac{1}{6} y &=1 \\\\-3 x-2 y &=0 \end{aligned}\right.\)

Step-by-Step Solution

Answer

The system of equations has no solution, which is proved by the process of elimination and confirmed graphically due to the fact that two lines never intersect.

Step 1: Preparation

Multiply the first equation by 12 and the second equation by 1 to obtain whole numbers and ease the process of elimination: \(12) * \(\frac{1}{4} x + \frac{1}{6} y = 1\) and \(1) * \(-3x - 2y = 0\), which results in: \[3x + 2y = 12\] and \[-3x - 2y = 0\].

Step 2: Elimination

We add both equations to get \[3x - 3x + 2y - 2y = 12 + 0\], which simplifies as \[0 = 12\]. Since 0 is not equal to 12, the system of equations has no solution.

Step 3: Graphical Verification

Plot the lines of the two equations in an XY plane. Since the two lines are parallel to each other, they never intersect. This confirms that the system of equations has no solution.

Key Concepts

Understanding the Elimination Method in Solving Systems of Equations

Solving a system of equations using the elimination method is like finding a common point of intersection for two lines represented by the equations. The goal is to eliminate one of the variables by adding or subtracting the equations from each other. Here’s how it works:

• First, adjust the equations so they are in a simple form where variables can easily cancel each other. This usually involves multiplying one or both of the equations by a number to align coefficients.
• For the system \[\begin{align*}\frac{1}{4}x + \frac{1}{6}y &= 1 \-3x - 2y &= 0\end{align*}\]we can multiply the first equation by 12 to get whole numbers.
• After simplification, our new equations become \(3x + 2y = 12\) and \(-3x - 2y = 0\).
• Next, add the equations. Here, \(3x\) cancels \(-3x\) and \(2y\) cancels \(-2y\), leaving \(0 = 12\).
• Since this statement is false, it indicates that the system has no solution.

The elimination method is a reliable technique, especially when the system has an obvious pair of variables to cancel out.

Parallel Lines and Their Role in Systems of Equations

In the context of linear equations, understanding the concept of parallel lines is crucial. Two lines are parallel if they have the same slope but different y-intercepts. This means they never meet.

• For instance, in our system, both transformed equations, \(3x + 2y = 12\) and \(-3x - 2y = 0\), were checked for parallelism.
• If you rearrange these into slope-intercept form \(y = mx + b\), you’ll find both have the same slope \(m\).
• This results in parallel lines on the graph.
• Parallel lines indicate inconsistency between the equations, which translates to no possible solutions because no common intersection point (solution) exists.

Grasping the concept of parallel lines helps elucidate why certain systems of equations lack solutions entirely.

Using a Graphing Utility to Visualize Solutions

Graphing utilities are invaluable tools for understanding systems of equations, providing a visual confirmation of algebraic solutions. Employing a graphing utility involves:

• Inputting each equation into the utility to plot them on an XY-plane.
• For instance, graphing \(3x + 2y = 12\) and \(-3x - 2y = 0\) will show two distinct lines.
• The graph distinctly shows whether lines intersect at any point, hence indicating possible solutions, or if they are parallel, showing no solutions.
• In this problem, the graph verifies the algebraic solution. The plotted parallel lines visually affirm no intersection, confirming no solutions.

By using a graphing utility, it becomes easier to see the practical outcome of theoretical solutions, making abstract concepts more tangible.