Problem 40 - 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. $$ \left\\{\begin{array}{l} \frac{x}{6}-\frac{y}{2}=\frac{1}{3} \\ x+2 y=-3 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution set for the system of equations is \(\{(x, y) : x = -1, y = -1\}\)

Step 1: Simplify the Equations

Simplify the first equation by multiplying the entire equation by 6 to clear the fractions. This gives \(x - 3y = 2\). Keep the second equation as it is, \(x+2y = -3\). So, our new system is \(\left\{ \begin{array}{l} x - 3y = 2 \ x+2y = -3 \end{array} \right.\)

Step 2: Solve for x

To solve for \(x\) from the two equations, we add the two equations together. This gives \((x - 3y) + (x + 2y) = 2 - 3\). After simplifying, we find \(2x - y = -1\), and \(x = 0.5y - 0.5\)

Step 3: Substitute x into second equation

Substitute \(x\) in terms of \(y\) into the second equation. This gives \((0.5y - 0.5) + 2y = -3\). Solving for \(y\) we find that \(y = -1\)

Step 4: Solve for x using the value of y

Substitute \(y = -1\) back to the equation for \(x\). We get \(x = 0.5(-1) - 0.5 = -1\)

Step 5: Express the solution set in set notation

The solution set is \(\{(x, y) : x = -1, y = -1\}\).

Key Concepts

Understanding Systems with No Solution

A system of linear equations might have no solution if the lines represented by the equations are parallel. Parallel lines never intersect, so there are no points that satisfy both equations simultaneously. Here's how you can identify such a system:

• Compare the slopes of the equations. If the slopes are the same but the lines have different y-intercepts, the system has no solution.
• This results in an inconsistency when trying to solve the system, like ending with a false statement such as \(0 = 5\).

These are cases where the equations are contradictory, and no (x, y) pair satisfies both conditions.

Exploring Systems with Infinitely Many Solutions

Sometimes, a system of equations will essentially be the same line. This happens when both equations represent the same line, resulting in infinitely many solutions because every point on the line is a solution. To recognize such a system:

• Check if the equations are multiples of each other. If they are, every solution of one equation is a solution to the other.
• When simplified, both equations wind up being identical, like \(x - 3y = 2\) and \(-2(x - 3y) = -4\).

In this scenario, your solution set could be expressed in set notation as all points (x, y) that lie on the identical line.

Using Set Notation for Solution Sets

Set notation is a concise way to express the solutions of a system of equations. It's especially useful when dealing with systems with no solution or infinitely many solutions. Here's a brief intro on how to use it:

• For a specific solution like we saw in the exercise, you write it as \(\{(x, y) : x = -1, y = -1\}\), which states the exact values of x and y that satisfy the equations.
• If there are no solutions, you might see it expressed as \(\emptyset\) or \(\{ \} \), indicating the empty set, meaning no solutions exist.
• For infinitely many solutions, you might use set-builder notation such as \(\{(x, y) : \text{equation representing the line}\}\), to describe all points on the line.

Understanding these notations helps you clearly communicate the nature of a solution set, whether it's specific, non-existent, or infinite.