Problem 34 - College Algebra

Question

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. \(\left\\{\begin{array}{l}9 x-3 y-12 \\ y-3 x-4\end{array}\right.\)

Step-by-Step Solution

Answer

The given system of equations has no solution, represented by set notation as \(\emptyset\).

Step 1: Rearrange the equations

Before attempting to solve the system, rearrange both equations when necessary to have the same variables on the left side. This is done so that it's easier to compare or add them together. The system then becomes:\(9x−3y=12\) => Equation (1)\(3x - y = -4\) => Equation (2)

Step 2: Solving the system

The equations can be solved using the method of substitution or elimination. Since both equations have the same terms but with different coefficients on the left side (-3y and -y, 9x and 3x), we could use the method of elimination. Multiply equation (2) by 3 to make the coefficients of y the same:\(9x - 3y = 12\) => Equation (1)\(9x - 3y = -12\) => New Equation (2)

Step 3: Follow ups of the system

After performing the operations, we have two identical equations on the left side, but the right sides are different. This means the system has no solution because the two lines represented by the equations are parallel lines and they will never intersect. This can also be verified by setting the two equations equal to each other like 12=-12, which is not a true statement, meaning there is no solution.

Step 4: Express the solution in set notation

In set notation, no solution is represented by an empty set, denoted by \(\emptyset\).

Key Concepts

Elimination Method

The elimination method is a powerful approach for solving a system of linear equations. It works by systematically eliminating one of the variables by adding or subtracting the equations. This simplifies the system to a single equation with one unknown, making it easier to solve.

Here is how you can use the elimination method step-by-step:

• First, rearrange both equations so that the variables align perfectly when written in an equation set.
• Next, look at the coefficients of the variables in both equations. Multiply one or both equations by a number that will make the coefficients of one of the variables equal in magnitude and opposite in sign.
• Now, add or subtract the equations to eliminate one variable.
• Solve the resulting equation for the remaining variable.
• Finally, substitute this value back into one of the original equations to find the value of the other variable.

This method is straightforward and efficient, especially when the equations can be easily manipulated to cancel out a variable. However, be careful when the system leads to unusual results, like discovering two equations define parallel lines. This can indicate that the system might have no solution.

Set Notation

Set notation is a mathematical tool for expressing the solution set of an equation or system of equations clearly and concisely. It helps us to avoid ambiguity when defining the solutions or lack thereof.

Set notation uses a combination of curly brackets and specific symbols to denote the nature of the solution:

• An empty set, represented by the symbol \( \emptyset \), indicates that there is no solution to the system of equations. This means no values for the variables meet all the conditions simultaneously.
• A set with elements, for example \( \{ x, y \} \), lists all possible solutions for the variables.

Using set notation provides a clear and formal way to represent solutions, helping to communicate mathematical findings accurately. For systems that don't intersect at all—meaning they have no solution—we use the empty set. For solutions that cover a range or are infinite, set notation can explicitly state these relationships.

Parallel Lines

Parallel lines play a significant role in understanding the solutions of a system of equations. When we have a graph of two linear equations, the lines may either intersect, be parallel, or coincide.

Two linear equations represent parallel lines if they have the same slope but different intercepts. This results in no points of intersection, indicating there is no common solution. The slopes and intercepts can be found by rearranging each equation into the slope-intercept form, \( y = mx + b \):

• If the slopes \( m_1 \) and \( m_2 \) are equal and \( b_1 \) and \( b_2 \) (intercepts) are not equal, the lines are parallel.

Understanding parallel lines in the context of a system of equations helps us quickly identify when a system has no solution, as these lines will never meet no matter the values chosen for the variables.

No Solution Systems

No solution systems occur in linear equation sets when the lines representing the equations are parallel. This means they will never intersect, as they have identical slopes but different intercepts.

This scenario arises under very specific circumstances:

• Both equations in the system have the same slopes, thus moving in parallel directionally through the coordinate plane.
• The equations have different intercepts, meaning they are shifted vertically and on separate paths that do not cross.

For such systems, it is impossible to find a single pair of \( x \) and \( y \) that satisfies both equations simultaneously. Hence, the solution is an empty set, denoted by \( \emptyset \), indicating that no solutions exist. Recognizing no solution systems quickly can save time during problem-solving and help avoid unnecessary calculations.