Problem 18 - Precalculus: Mathematics for Calculus

Question

$$\left\\{\begin{aligned} 2 x+6 y &=0 \\ -3 x-9 y &=18 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system has no solution due to a contradiction in the simplified equations.

Step 1: Simplify the Equations

The given system can be simplified by dividing entire equations by any common factors. For Equation 1, divide every term by 2: \( x + 3y = 0 \). For Equation 2, divide every term by -3: \( x + 3y = -6 \). The new system becomes: \(\begin{aligned} x + 3y &= 0 \ x + 3y &= -6 \end{aligned}\).

Step 2: Examine the Simplified System

Notice that the two simplified equations, \(x + 3y = 0\) and \(x + 3y = -6\), contradict each other. This indicates that the system does not have a solution.

Step 3: Conclusion

Since the simplified equations result in a contradiction, the original system has no solution. The lines represented by the equations are parallel and never intersect.

Key Concepts

Linear Equations

Linear equations are the building blocks of algebra and are essential for understanding systems of equations. Varied in form, linear equations usually appear as straight lines when plotted on a graph. The basic form of a linear equation in two variables can be written as:

• \(ax + by = c\)

Here, \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. Linear equations suggest a constant rate of change and have no exponents higher than 1.
In the original exercise, we're dealing with two linear equations. Each of these tells us something about the relationship between \(x\) and \(y\). When solving such a system, we aim to find values of \(x\) and \(y\) that satisfy both equations simultaneously.

Parallel Lines

Parallel lines are a significant concept when dealing with linear equations. They occur when two lines have the same slope but different y-intercepts. Simply put, they will never meet, like the tracks of a railroad.For the equations \(x + 3y = 0\) and \(x + 3y = -6\), the left side of both equations \(x + 3y\) is identical, which indicates they have the same slope. However, their right sides differ: 0 and -6, respectively.

• Identical slopes mean they are parallel.
• Because they are parallel, they cannot intersect.

This is why the solution of this system of equations is classified as having no solution.
Graphically, these two lines will be horizontally shifted versions of each other, forever reaching towards infinity without crossing each other's path.

Inconsistent System

An inconsistent system of equations occurs when there are no solutions that satisfy all equations in the system simultaneously. This often arises when dealing with parallel lines, as shown in our given system of linear equations. After simplifying, we end up with the equations \(x + 3y = 0\) and \(x + 3y = -6\). These two contradictory statements imply that no point exists where the two equations are true at the same time. An inconsistent system is characterized by:

• Parallel lines that never intersect.
• Conflicting equations after simplification.

When faced with an inconsistent system, it’s crucial to recognize the signs—identical slopes but differing y-intercepts. Such a system mirrors a real-life scenario where two sets of conditions or rules can never align.