Problem 37 - College Algebra Essentials

Question

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. $$ \left\\{\begin{array}{r} x+3 y=2 \\ 3 x+9 y=6 \end{array}\right. $$

Step-by-Step Solution

Answer

The system of equations has an infinite number of solutions, which can be represented in set notation as: \(\{(x, y) : x + 3y = 2\}\).

Step 1: Simplification

First, simplify the equations into an easier form if possible. After dividing the second equation by 3, the two equations become: \[\begin{array}{r}x+3y=2\x+3y=2\end{array}\]

Step 2: Comparison of the equations

As a result, we have two identical equations. This indicates that these two equations would exactly overlap if we graph them. Therefore, there's a multitude of points on which the two equations coincide.

Step 3: Identifying solutions

So, any (x, y) that satisfies the equation \(x+3y=2\) is a solution to the whole system. Therefore, the system has an infinite number of solutions. In set notation, they can be represented as \(\{(x, y) : x + 3y = 2\}\).

Key Concepts

Infinite Solutions

When solving a linear system, you may encounter a situation where there are infinite solutions. This happens when the equations in the system represent the same line. Imagine two lines on a graph where one lies perfectly on top of the other. In such cases, every point that lies on the line is a solution to the system since both lines share every possible point.

For example, in the given system:

• Equation 1: \(x + 3y = 2\)
• Equation 2: \(3x + 9y = 6\)

After simplifying Equation 2 by dividing every term by 3, we get the same equation as Equation 1. Here, the two equations are identical, leading to infinite solutions since they share all points in common.This can simplify your problem-solving process by recognizing that you don’t need to solve further once you've established equivalence between the equations.

Identical Equations

Identical equations are at the heart of understanding when a system has infinite solutions. When you simplify the equations in a system and find that they are identical, this means you're dealing with the same line graphed twice. Identical equations mean each equation will yield the same answers for any values of \(x\) and \(y\) that satisfy them.

In the exercise, simplifying the second equation:

• Original: \(3x + 9y = 6\)
• Dividing by 3: \(x + 3y = 2\)

After this step, both equations are identical, confirming that the system doesn't just intersect at one point, but overlaps completely. Anytime you observe identical equations after simplifying, it tells you immediately that there are infinitely many solutions.

Graphing Systems

Graphing systems of linear equations is a powerful visual method to interpret solutions. When graphed, each equation will typically appear as a straight line on the coordinate plane. The solution is found where these lines intersect. However, with identical equations, as in our case, you’ll observe that the lines lie exactly on top of one another.

To graph the system:

• Plot \(x + 3y = 2\)
• Since the second equation simplifies to the same line, it overlaps entirely.

Rather than finding a single intersection point, the entire line is the solution set, giving us infinite solutions. This visual cue can help confirm your algebraic solution.

Set Notation

Set notation is a convenient way to express solutions to a system of equations, especially when dealing with infinite solutions. It allows you to succinctly describe all solutions simultaneously. Rather than listing every point that solves the system, which is impossible in the case of infinite solutions, set notation encapsulates them all efficiently.

For the given system, once simplified, the solution set is written using set notation:

• \(\{(x, y) : x + 3y = 2\}\)

This notation reads, "the set of all \((x, y)\) such that \(x + 3y = 2\)." It implies every ordered pair \((x, y)\) lying on the line is a solution, emphasizing the complete overlap of solutions between the identical equations. Set notation is not only efficient but also mathematically elegant for expressing an infinite set of solutions.