Question
define Solution of a system as an ordered pair
Answers
GPT-4o mini
In the context of a system of equations, particularly in the case of a system of two linear equations in two variables (typically \(x\) and \(y\)), a **solution** can be defined as an ordered pair \((x, y)\) that satisfies both equations simultaneously.
For example, consider the following system of linear equations:
1. \(2x + 3y = 6\)
2. \(x - y = 1\)
A solution to this system would be an ordered pair \((x, y)\) such that, when you substitute \(x\) and \(y\) into both equations, both equations hold true.
If we find the ordered pair \((2, 0)\):
1. Substituting into the first equation:
\[2(2) + 3(0) = 4 + 0 = 4 \quad (\text{not a solution})\]
2. Substituting into the second equation:
\[2 - 0 = 2 \quad (\text{not a solution})\]
Thus, \((2, 0)\) is not a solution.
Imagine we found \((3, 0)\) instead:
1. Substituting into the first equation:
\[2(3) + 3(0) = 6 + 0 = 6 \quad (\text{solution})\]
2. Substituting into the second equation:
\[3 - 0 = 3 \quad (\text{solution})\]
In this case, \((3, 0)\) is a valid solution because it satisfies both equations.
Therefore, a **solution of a system** can be defined as an ordered pair \((x, y)\) such that:
- It satisfies the first equation of the system.
- It satisfies the second equation of the system.
If the system has multiple solutions, it will have multiple ordered pairs. If no such pairs exist, the system is said to have no solution. If the equations represent the same line, there are infinitely many solutions (infinitely many ordered pairs).
For example, consider the following system of linear equations:
1. \(2x + 3y = 6\)
2. \(x - y = 1\)
A solution to this system would be an ordered pair \((x, y)\) such that, when you substitute \(x\) and \(y\) into both equations, both equations hold true.
If we find the ordered pair \((2, 0)\):
1. Substituting into the first equation:
\[2(2) + 3(0) = 4 + 0 = 4 \quad (\text{not a solution})\]
2. Substituting into the second equation:
\[2 - 0 = 2 \quad (\text{not a solution})\]
Thus, \((2, 0)\) is not a solution.
Imagine we found \((3, 0)\) instead:
1. Substituting into the first equation:
\[2(3) + 3(0) = 6 + 0 = 6 \quad (\text{solution})\]
2. Substituting into the second equation:
\[3 - 0 = 3 \quad (\text{solution})\]
In this case, \((3, 0)\) is a valid solution because it satisfies both equations.
Therefore, a **solution of a system** can be defined as an ordered pair \((x, y)\) such that:
- It satisfies the first equation of the system.
- It satisfies the second equation of the system.
If the system has multiple solutions, it will have multiple ordered pairs. If no such pairs exist, the system is said to have no solution. If the equations represent the same line, there are infinitely many solutions (infinitely many ordered pairs).
uhm short 1-2 sentence definition please
GPT-4o mini
A **solution of a system of equations** is an ordered pair \((x, y)\) that satisfies all equations in the system simultaneously. In the case of a system of two linear equations, it represents the point of intersection of the lines represented by the equations.