Problem 24 - Algebra and Trigonometry

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. $$\left\\{\begin{aligned}2 x-3 y&=-8 \\ 14 x-21 &y=3 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system has no solution as the equations represent parallel lines.

Step 1: Check Consistency Using Multiples

To determine if the system is consistent, look for proportionality between the equations. The second equation is \(14x - 21y = 3\). Notice that if you multiply the first equation \(2x - 3y = -8\) by 7, it becomes \(14x - 21y = -56\). These two equations have different constant terms on the right side (-56 and 3), so they are not multiples of each other while having the same left-hand side.

Step 2: Identify System Type

Since the equations are proportional in terms of variables but not in terms of constants, the system of equations represents two parallel lines that do not intersect. This means the system has no solutions.

Key Concepts

Consistency of Linear Systems

When we're dealing with a system of linear equations, one key question is whether the system is consistent or inconsistent. A consistent system has at least one solution, while an inconsistent one has none. To determine consistency, we need to explore the relationships between the equations involved.

One approach is to see if the equations are multiples of each other. If the entire equation, including the coefficients of both variables and the constant, can be scaled to match another equation exactly, the system is consistent. If only the variables are proportional but the constants are not, the system is inconsistent, often indicating parallel lines that don't intersect.

• A consistent system: Equations are proportional in entirety, or intersect at least at one point.
• An inconsistent system: Equations are proportional in part but differ in constants, indicating no intersection.

Understanding these differences can help solve or determine the nature of a system swiftly.

Proportionality of Equations

Proportionality tells us a lot about the relationship between two equations in a system. If the coefficients of the variables are in the same ratio for both equations, then these variables are considered to be proportional.

In math, we often express this as one equation being a scalar multiple of the other. Take the equations:

• First equation: \(2x - 3y = -8\)
• Second equation: \(14x - 21y = 3\)

By comparing coefficients, we can see the left sides are multiples of each other. However, the key is to find out if this holds for the constant too. If not (which is the case here as -56 is not equal to 3 when we multiply the first equation by 7), the proportionality does not span the entire equation.

• Only having proportional coefficients means possible parallelism.
• Full proportionality (including constants) means overlap or identical lines.

Proportionality in systems provides clues on the nature of solutions: either infinite, one, or none.

Parallel Lines in Algebra

Parallel lines play an integral role in understanding the solutions of linear systems. Two lines are parallel if they share the same slope yet never intersect. This can be seen visually by plotting the lines, or algebraically by assessing the equations.

Mathematically, if two equations have proportional coefficients for both variables but differ in their constant terms, they represent parallel lines. The example equations \(2x - 3y = -8\) and \(14x - 21y = 3\) confirm this, as we've established that the variable proportions match but the constants do not. Consequently, their graphical representation would show lines that run side-by-side indefinitely without meeting.

• Parallel nature comes from equal ratios in linear terms and differing constants.
• Lack of intersection implies no common solutions; hence, the system is inconsistent.

By identifying parallel lines in equations, we recognize systems with no intersection points or solutions.