System B x=2y=6 =x+2y=-6 The system has no solution. O The system has a unique solution: (x, y) = ( The system has infinitely many solutions. They must satisfy the following equation: y = 6 + 2y
System B x=2y=6 =x+2y=-6 The system has no solution. O The system has a unique solution: (x, y) = ( The system has infinitely many solutions. They must satisfy the following equation: y = 6 + 2y
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter7: Conic Sections And Quadratic Systems
Section7.4: Solving Nonlinear Systems Of Equations
Problem 63E
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Question
![**System of Equations: Analysis and Solution Types**
Consider the following system of equations, referred to as System B:
\[
\begin{cases}
x - 2y = 6 \\
-x + 2y = -6
\end{cases}
\]
To determine the nature of the solutions for this system, we need to analyze the equations provided.
**Options for System B:**
1. **The system has no solution.**
2. **The system has a unique solution.**
3. **The system has infinitely many solutions. They must satisfy the following equation:**
\[
(x, y) = \boxed{\ }
\]
\[
y = \boxed{\frac{6 + 2}{2}}
\]
**Detailed Explanation:**
To understand the type of solution this system has, let’s first simplify the given equations.
Starting with:
\[
x - 2y = 6
\]
And:
\[
-x + 2y = -6
\]
If we add these two equations together:
\[
(x - 2y) + (-x + 2y) = 6 + (-6)
\]
Simplifying the left side gives us:
\[
0 = 0
\]
The resultant equation, \(0 = 0\), indicates that the two original equations are dependent and represent the same line. This means that the system has infinitely many solutions, and any point \((x, y)\) on this line will satisfy both equations.
Thus, the choice is:
**The system has infinitely many solutions.**
They satisfy the relation:
\[
y = \boxed{\text{6-2}}
\]
In conclusion, for the provided System B, the solution is infinitely numerous, meaning that any pair \((x, y)\) that lies on the line represented by the equations will be a valid solution.
**Note**: The text contains input fields labeled as: \( \boxed{\ } \) where further specific calculations or solutions could be entered based on user input or additional context.
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Transcribed Image Text:**System of Equations: Analysis and Solution Types**
Consider the following system of equations, referred to as System B:
\[
\begin{cases}
x - 2y = 6 \\
-x + 2y = -6
\end{cases}
\]
To determine the nature of the solutions for this system, we need to analyze the equations provided.
**Options for System B:**
1. **The system has no solution.**
2. **The system has a unique solution.**
3. **The system has infinitely many solutions. They must satisfy the following equation:**
\[
(x, y) = \boxed{\ }
\]
\[
y = \boxed{\frac{6 + 2}{2}}
\]
**Detailed Explanation:**
To understand the type of solution this system has, let’s first simplify the given equations.
Starting with:
\[
x - 2y = 6
\]
And:
\[
-x + 2y = -6
\]
If we add these two equations together:
\[
(x - 2y) + (-x + 2y) = 6 + (-6)
\]
Simplifying the left side gives us:
\[
0 = 0
\]
The resultant equation, \(0 = 0\), indicates that the two original equations are dependent and represent the same line. This means that the system has infinitely many solutions, and any point \((x, y)\) on this line will satisfy both equations.
Thus, the choice is:
**The system has infinitely many solutions.**
They satisfy the relation:
\[
y = \boxed{\text{6-2}}
\]
In conclusion, for the provided System B, the solution is infinitely numerous, meaning that any pair \((x, y)\) that lies on the line represented by the equations will be a valid solution.
**Note**: The text contains input fields labeled as: \( \boxed{\ } \) where further specific calculations or solutions could be entered based on user input or additional context.
**Navigation Buttons:**
1. **Previous**
2. **Next**
3. **Example**
These buttons allow the user to navigate through the content easily, helping in reviewing or learning detailed material as required.
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