Question
Which of the following systems of winning equations has no solution for x - y = 17 4x + y = 3 4 x - y = - 17 for x - y = 3 4x + y = 17 for x minus y equals 3 4x - y = 17 4x - y equals 3
Answers
Answered by
GPT-4o mini
To determine which of the provided systems of equations has no solution, we can analyze each system of equations to see if they are consistent or inconsistent.
### System 1
1. \(x - y = 17\)
2. \(4x + y = 3\)
3. \(4x - y = -17\)
Let's analyze this system:
1. From \(x - y = 17\), we can express \(y\) in terms of \(x\):
\[
y = x - 17
\]
2. Substituting \(y\) into the second equation:
\[
4x + (x - 17) = 3 \implies 5x - 17 = 3 \implies 5x = 20 \implies x = 4
\]
Then \(y = 4 - 17 = -13\).
3. Now, substituting \(x = 4\) and \(y = -13\) into the third equation:
\[
4(4) - (-13) = 16 + 13 = 29 \neq -17
\]
So the third equation is inconsistent with the first two. Therefore, this system has no solution.
### System 2
1. \(x - y = 3\)
2. \(4x + y = 17\)
From the first equation:
\[
y = x - 3
\]
Substituting \(y\) into the second equation:
\[
4x + (x - 3) = 17 \implies 5x - 3 = 17 \implies 5x = 20 \implies x = 4
\]
Then \(y = 4 - 3 = 1\). Both equations are satisfied by \(x = 4, y = 1\).
### System 3
1. \(x - y = 3\)
2. \(4x - y = 17\)
From the first equation:
\[
y = x - 3
\]
Substituting \(y\) into the second equation:
\[
4x - (x - 3) = 17 \implies 4x - x + 3 = 17 \implies 3x + 3 = 17 \implies 3x = 14 \implies x = \frac{14}{3}
\]
Then, \(y = \frac{14}{3} - 3 = \frac{14}{3} - \frac{9}{3} = \frac{5}{3}\). Both equations are satisfied.
### System 4
1. \(4x - y = 17\)
2. \(4x - y = 3\)
These two equations represent parallel lines because they have the same left side but different constants on the right side. Therefore, there is no solution for this system.
### Conclusion
The systems of equations with no solutions are the first and the last systems. If only one system is required, the last one, which has inconsistent equations, is a clear example of a system with no solution.
**Final Answer: The last system has no solution, as it consists of two parallel lines.**
### System 1
1. \(x - y = 17\)
2. \(4x + y = 3\)
3. \(4x - y = -17\)
Let's analyze this system:
1. From \(x - y = 17\), we can express \(y\) in terms of \(x\):
\[
y = x - 17
\]
2. Substituting \(y\) into the second equation:
\[
4x + (x - 17) = 3 \implies 5x - 17 = 3 \implies 5x = 20 \implies x = 4
\]
Then \(y = 4 - 17 = -13\).
3. Now, substituting \(x = 4\) and \(y = -13\) into the third equation:
\[
4(4) - (-13) = 16 + 13 = 29 \neq -17
\]
So the third equation is inconsistent with the first two. Therefore, this system has no solution.
### System 2
1. \(x - y = 3\)
2. \(4x + y = 17\)
From the first equation:
\[
y = x - 3
\]
Substituting \(y\) into the second equation:
\[
4x + (x - 3) = 17 \implies 5x - 3 = 17 \implies 5x = 20 \implies x = 4
\]
Then \(y = 4 - 3 = 1\). Both equations are satisfied by \(x = 4, y = 1\).
### System 3
1. \(x - y = 3\)
2. \(4x - y = 17\)
From the first equation:
\[
y = x - 3
\]
Substituting \(y\) into the second equation:
\[
4x - (x - 3) = 17 \implies 4x - x + 3 = 17 \implies 3x + 3 = 17 \implies 3x = 14 \implies x = \frac{14}{3}
\]
Then, \(y = \frac{14}{3} - 3 = \frac{14}{3} - \frac{9}{3} = \frac{5}{3}\). Both equations are satisfied.
### System 4
1. \(4x - y = 17\)
2. \(4x - y = 3\)
These two equations represent parallel lines because they have the same left side but different constants on the right side. Therefore, there is no solution for this system.
### Conclusion
The systems of equations with no solutions are the first and the last systems. If only one system is required, the last one, which has inconsistent equations, is a clear example of a system with no solution.
**Final Answer: The last system has no solution, as it consists of two parallel lines.**