Problem 34 - Algebra and Trigonometry

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 and can be expressed in set notation as {(x, y) | y = 3x - 4}. Any value of x and the corresponding value of y that satisfies y = 3x - 4 is a solution to this system of equations.

Step 1: Simplify the equations

Given two equations, 9x - 3y = 12 and y = 3x - 4. We can simplify the first equation by dividing by 3 to reduce coefficients, then becomes 3x - y = 4.

Step 2: Substitution or Elimination

Substitute y from the second equation to the first. This means wherever y appears, replace it with 3x - 4. The first equation (3x - y = 4) becomes 3x - (3x - 4) = 4. Simplifying this gives 4 = 4.

Step 3: Determine the kind of system

The result after simplification in step 2 is a true statement which implies that the original system of equations has infinitely many solutions. This means that any value of x and corresponding value of y is a solution.

Step 4: Express solution set in set notation

Expressing the solution in set notation will involve showing the relationships of the values of the variables. In this case, the solution set is {(x, y) | y = 3x - 4}. This implies that all pairs of (x, y) where y is equal to 3x - 4 are solutions to the system.

Key Concepts

Method Of Substitution

The method of substitution is a powerful technique for solving systems of equations. It involves replacing one variable with an expression derived from another equation in the system. Here's how it works in simple steps:

• First, solve one of the equations for one of its variables. In our exercise, the second equation is already solved for y, which is perfect for substitution.
• Next, you'll substitute the expression for this variable into the other equation. This is done by replacing the variable with its equivalent expression from the other equation.
• After substitution, you will have an equation with a single variable. Solve this equation to find the value of that variable.
• Finally, substitute the value you found back into the equation you first used to solve for the other variable.

In the case of our exercise, since the simplification led to a true statement without giving a specific value for x, it indicates that we have infinitely many solutions, not just one. This is a special case in substitution where one equation is a multiple of the other.

Identifying System Solutions

When solving systems of equations, identifying the kind of solutions a system has is crucial. There are three potential outcomes:

• A unique solution: This is when the system has exactly one solution pair, meaning one value for x and one value for y that satisfy all equations.
• No solution: This occurs when there is no pair that satisfies all equations in the system, often because the lines are parallel and never intersect.
• Infinitely many solutions: This happens when the equations represent the same line, hence every point on the line is a solution.

To identify the solution, we look at the simplified form of the equations after applying substitution or elimination. Our exercise resulted in an equation that holds true for all values of x, thus indicating infinitely many solutions. If the simplification leads to a contradiction, such as a statement that is always false (for example, 0 = 5), then the system has no solution.

Set Notation

Set notation is a succinct way to express a collection of objects, or in the context of algebra, the solutions set to a system of equations. Here's a quick primer on how to read and write set notation:

• The curly braces \( \{ \}\) represent a set of elements.
• An element within a set can be a number, a pair, or any mathematical object.
• A vertical bar or colon \( | \) or \( : \) signifies 'such that.' This precedes the condition that members of the set must fulfill.
• In set notation, for system solutions, we list the variable pair and then describe their relationship. For example, \(\{(x,y) | y = 3x - 4\}\), expresses all pairs (x, y) that satisfy the equation y = 3x - 4.

In the exercise, the solution set is expressed as all pairs where y is substituted as 3x - 4, meaning any (x, y) that keeps this relationship intact is part of the solution set.

System Of Linear Equations

A system of linear equations consists of two or more equations with the same set of variables. The goal is to find the values of these variables that satisfy all equations simultaneously. Characteristics of a system of linear equations include:

• Linear equations, which graph as straight lines.
• One or more variables, but each term is to the first power.
• Consistent systems have at least one solution pair, while inconsistent systems have none.

In our exercise, the system is consistent and dependent, as the equations represent the same line, yielding infinitely many solutions. It is important that each step in solving such systems, be it through substitution, elimination, or graphical methods, maintains the equality and integrity of the original equations to correctly identify the nature of the system's solutions.