Problem 40 - Thinking Mathematically

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 system of equations has infinitely many solutions represented as \(x \epsilon R\) and \(y \epsilon R\), meaning both x and y can be any real number.

Step 1: Substitute

Charge the value of y from the second equation into the first equation. This gives: \[9x - 3(3x - 4) = 12.\]

Step 2: Simplify

Simplify the equation by distributing -3 into \(3x - 4\). This will yield the equation: \[9x - 9x +12 = 12.\]

Step 3: Solve for x

From the above, the x values cancel out and you're left with: \[12 = 12.\] This implies that x can hold any real value.

Step 4: Express the solution set in set notation

The solution set for x will be the set of all real numbers, represented in set notation as: \(x \epsilon R.\)

Step 5: Evaluate y

From the second equation, and given that x can be any real value, it is evident that y will also be any real number. We can write: \(y = 3x - 4.\)

Step 6: Express the solution set for y in set notation

The solution set for y will be the set of all real numbers, represented in set notation as: \(y \epsilon R.\)

Key Concepts

Method of Substitution

The method of substitution is a fundamental technique used in solving systems of linear equations. This technique involves replacing one variable in an equation with an equivalent expression from another equation in the system.

Here's how to use the method of substitution in a step-by-step format:

• First, solve one of the equations for one variable in terms of the others. In our example, the second equation is already solved for y: \( y = 3x - 4 \).
• Next, substitute this expression into the other equation(s) in place of y. Here, we substitute \( y = 3x - 4 \) into the first equation: \( 9x - 3y = 12 \).
• Simplify the resulting equation and solve for the remaining variable(s). Once simplified, if the variable disappears and you get a true statement, like \( 12=12 \), it indicates that there are many solutions to the system — in fact, an infinite number.

It is crucial to perform each step correctly, to ensure the solution is accurate and makes sense within the context of the problem.

Infinite Solutions

In the context of systems of equations, the term 'infinite solutions' refers to a scenario where there are countless solutions that satisfy all the equations in the system. This usually happens in systems where the equations are not independent, meaning they essentially express the same relationship in different forms.

One clear sign of infinite solutions is when, after simplifying the equations, all variables cancel out and you're left with a true statement, such as \( 12=12 \). It indicates that every pair of x and y that satisfies one equation will also satisfy the other. This results in a scenario where the graph of both equations would show two identical lines, thereby overlapping entirely, and every point on the line is a solution to the system.

Set Notation

Set notation is a way to succinctly describe a collection of objects, numbers, or other entities. In the case of solutions to equations, we often use set notation to express the solution set.

The following symbols are commonly used in set notation:

• The capital letter 'R' denotes the set of all real numbers.
• The symbol '\(\epsilon\)' means 'is an element of,' so writing \(x \epsilon R\) indicates that the variable x is any real number.
• Curly braces '{' and '}' encapsulate the members of the set, or a description of its members.

When we have infinite solutions, we use set notation to imply that the solutions encompass all real numbers. For example, when both x and y have all real numbers as their solutions, we would write their solution sets as \(x \epsilon R\) and \(y \epsilon R\), respectively.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. It provides the language and framework for solving systems of equations, like the one in our example.

In linear algebra:

• Each linear equation in a system can be represented as a line on a graph.
• A solution to a system of linear equations corresponds to a point where the lines intersect.
• If lines coincide, the system has infinite solutions—every point on the line is a valid solution.
• If lines are parallel and never intersect, the system has no solution.

This mathematical discipline is not only essential for solving simple systems but also forms the foundation for more complex topics in mathematics and applied sciences, including computer graphics, optimization, and machine learning.