Problem 31 - College Algebra

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. \(x=9-2 y\) \(x+2 y=13\)

Step-by-Step Solution

Answer

The system of equations has no solution.

Step 1: Rearrange the equations

Rearrange the first equation into: \(x=9-2y\) which leads to \(x+2y=9\). The second equation is: \(x+2y=13\). We can see these two equations cannot be true at the same time unless \(9=13\).

Step 2: Compare the equations

Compare the two equations \(x+2y=9\) and \(x+2y=13\), we can see that while the left hand side of both equations are the same, the right hand side is different. This means that the given equation system has no solution.

Key Concepts

No Solution in Systems of Equations

In some systems of equations, no values exist that satisfy all equations simultaneously. These are known as systems with no solution. For example, when we rearrange the original system into two equations:

• First equation: \(x + 2y = 9\)
• Second equation: \(x + 2y = 13\)

We see that both equations contradict each other because they imply that \(9 = 13\), which is impossible. Hence, there is no intersection point for these equations when graphed, signifying no solution.
Systems with no solutions are known as inconsistent systems. Understanding this helps identify situations where equations define parallel lines that never meet.

Infinitely Many Solutions

While some systems have no solution, others can have infinitely many solutions. This occurs when the equations describe the same line. Every point on that line is a solution.
How can you tell if a system has infinitely many solutions? By transforming both equations into the same form or realizing one is a multiple of the other. This signifies they overlap completely, hence sharing every solution point.
For example, if two equations are:

• \(y = 2x + 3\)
• \(2y = 4x + 6\)

Solving and checking will show both are the same line, confirming infinitely many solutions.

Set Notation for Solution Sets

Set notation is a mathematical tool used to express solutions or groups of numbers precisely. It provides a clear and concise way to represent the solution set of equations.
For systems with no solution, the solution set is represented as \(\emptyset\), indicating an empty set. When there are infinitely many solutions, we describe it by writing a general solution using set-builder notation. For example, for an equation like:

• \(x = 2y + 5\)

The solution set could be expressed as \(\{ (x, y) | x = 2y + 5 \} \). This notation effectively communicates that all pairs \((x, y)\) matching the relationship are solutions.

Understanding Algebraic Expressions

Algebraic expressions form the backbone of systems of equations. They include variables, numbers, and operations like addition, subtraction, etc. Understanding them is crucial for solving equations efficiently.
For example, the expression \(x+2y=13\) consists of:

• Variables \(x\) and \(y\)
• Coefficients 2 and 1 for \(y\) and \(x\)
• Constants such as 13

Grasping these components enables the manipulation and rearrangement of equations to uncover solutions. It's like cracking a code - once you recognize the parts, solving becomes straightforward.
In systems of equations, recognizing like expressions speeds up the process of solving and finding solutions.