Problem 24 - Precalculus: Mathematical for Calculus

Question

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 3. $$\left\\{\begin{array}{cc} 2 x-3 y= & -8 \\ 14 x-21 y= & 3 \end{array}\right.$$

Step-by-Step Solution

Answer

The system has no solution; the equations are inconsistent parallel lines.

Step 1: Write Down the Equations

The system of equations is given as: 1) \(2x - 3y = -8\) 2) \(14x - 21y = 3\). We need to find the values of \(x\) and \(y\) or determine if there are no/many solutions.

Step 2: Simplify the Second Equation

Notice that the second equation \(14x - 21y = 3\) can be simplified. Divide the entire equation by 7:\(\frac{14x}{7} - \frac{21y}{7} = \frac{3}{7}\), which simplifies to \(2x - 3y = \frac{3}{7}\).

Step 3: Compare the Equations

Now, compare the two simplified equations:1) \(2x - 3y = -8\)2) \(2x - 3y = \frac{3}{7}\)The left sides of the equations are identical, but the right sides are not.

Step 4: Determine the Nature of the System

Since the left sides of the equations are the same, but the right sides are different, this means that the two equations represent parallel lines. Parallel lines never intersect, thus there is no solution.

Step 5: Conclusion

The system of equations is inconsistent because it represents two parallel lines, which means there is no solution to this system.

Key Concepts

Parallel Lines

Understanding the concept of parallel lines is crucial when solving systems of linear equations. In geometry, parallel lines are lines in a plane that never meet, no matter how far they extend. This is because they are perfectly aligned with each other at a constant distance apart and have the same slope.

When translating this into the world of algebra, if two linear equations in a system result in equations with identical left-side expressions (i.e., the same coefficients for both variables), but differing constant terms on the right side, they represent parallel lines. For example, in our original problem, both equations simplify to forms that have identical left sides:

• Equation 1: \( 2x - 3y = -8 \)
• Equation 2: \( 2x - 3y = \frac{3}{7} \)

Since these lines have the same slope of \( \frac{2}{3} \), they are parallel to one another and therefore will never intersect.

Inconsistent System

An inconsistent system of equations is a system that has no solutions. This happens when the equations represent parallel lines that do not meet at any point in the coordinate plane. In our example, after simplifying the second equation in the system, both equations ended up having the same left sides but different right sides:

• First equation: \( 2x - 3y = -8 \)
• Second equation: \( 2x - 3y = \frac{3}{7} \)

This difference means that there is no combination of \( x \) and \( y \) values that can satisfy both equations simultaneously.

When you come across a system where the equations align to create parallel lines, remember that this is a feature of an inconsistent system. It's important to recognize this pattern because attempting to solve such a system for intersection points would be futile since parallel lines never cross each other.

Infinitely Many Solutions

Problems that result in infinitely many solutions often involve coincident lines. These are different from parallel lines. Coincident lines lie exactly on top of each other as they are essentially the same line represented by different forms of the same equation.

To find such a scenario, when simplifying the given system of equations, both equations would end up identical after simplification. For a visual, imagine starting with different equations that simplify to:

• Equation 1: \( 2x - 3y = C \)
• Equation 2: \( 2x - 3y = C \)

For different constant \( C \). Thus, any point on this line is a solution, leading to infinitely many solutions because there are infinite points on a line.

In the original exercise, the system did not have infinitely many solutions because the right sides were different after simplification, leading to the case of parallel lines instead. Always look closely at how the simplified equations compare to discern between these scenarios.