Answer :
To determine the solutions to the given system of linear equations, let's analyze the pairs of equations step-by-step.
### Equations in the system:
1. [tex]\( x + y = -3 \)[/tex]
2. [tex]\( x + y = 4 \)[/tex]
3. [tex]\( -x - y = 3 \)[/tex]
4. [tex]\( x + y = 7 \)[/tex]
### Step-by-step Analysis and Solutions:
#### Pair 1: [tex]\( x + y = -3 \)[/tex] and [tex]\( x + y = 4 \)[/tex]
Let's compare these two equations:
[tex]\[ x + y = -3 \][/tex]
[tex]\[ x + y = 4 \][/tex]
If we subtract the first equation from the second equation:
[tex]\[ (x + y) - (x + y) = 4 - (-3) \implies 0 = 7 \][/tex]
This is a contradiction, which means there is no solution for this pair of equations.
#### Pair 2: [tex]\( x + y = -3 \)[/tex] and [tex]\( -x - y = 3 \)[/tex]
Let's compare these two equations:
[tex]\[ x + y = -3 \][/tex]
[tex]\[ -x - y = 3 \][/tex]
If we add these two equations together:
[tex]\[ (x + y) + (-x - y) = -3 + 3 \implies 0 = 0 \][/tex]
This indicates that the two equations are actually the same when simplified, i.e., they are dependent. Thus, they produce an infinitely many solutions defined by:
[tex]\[ y = -x - 3 \][/tex]
Any point [tex]\((x, -x - 3)\)[/tex] is a solution to these equations.
#### Pair 3: [tex]\( -x - y = 3 \)[/tex] and [tex]\( x + y = 7 \)[/tex]
Let's re-express and compare these equations:
[tex]\[ -x - y = 3 \][/tex]
[tex]\[ x + y = 7 \][/tex]
If we add these equations together:
[tex]\[ (-x - y) + (x + y) = 3 + 7 \implies 0 = 10 \][/tex]
This is another contradiction, which means there is no solution for this pair of equations.
### Summary of Solutions:
1. The pair [tex]\( x + y = -3 \)[/tex] and [tex]\( x + y = 4 \)[/tex] leads to no solution (contradiction).
2. The pair [tex]\( x + y = -3 \)[/tex] and [tex]\( -x - y = 3 \)[/tex] has infinitely many solutions with [tex]\( y = -x - 3 \)[/tex].
3. The pair [tex]\( -x - y = 3 \)[/tex] and [tex]\( x + y = 7 \)[/tex] leads to no solution (contradiction).
So, the summarized result for these systems of equations is:
- One solution: [tex]\( [] \)[/tex]
- No solution: [tex]\( [] \)[/tex]
- Infinitely many solutions defined by [tex]\( y = -x - 3 \)[/tex]
This analysis explains why the results are: no solution for pairs 1 and 3, and infinitely many solutions for pair 2.
### Equations in the system:
1. [tex]\( x + y = -3 \)[/tex]
2. [tex]\( x + y = 4 \)[/tex]
3. [tex]\( -x - y = 3 \)[/tex]
4. [tex]\( x + y = 7 \)[/tex]
### Step-by-step Analysis and Solutions:
#### Pair 1: [tex]\( x + y = -3 \)[/tex] and [tex]\( x + y = 4 \)[/tex]
Let's compare these two equations:
[tex]\[ x + y = -3 \][/tex]
[tex]\[ x + y = 4 \][/tex]
If we subtract the first equation from the second equation:
[tex]\[ (x + y) - (x + y) = 4 - (-3) \implies 0 = 7 \][/tex]
This is a contradiction, which means there is no solution for this pair of equations.
#### Pair 2: [tex]\( x + y = -3 \)[/tex] and [tex]\( -x - y = 3 \)[/tex]
Let's compare these two equations:
[tex]\[ x + y = -3 \][/tex]
[tex]\[ -x - y = 3 \][/tex]
If we add these two equations together:
[tex]\[ (x + y) + (-x - y) = -3 + 3 \implies 0 = 0 \][/tex]
This indicates that the two equations are actually the same when simplified, i.e., they are dependent. Thus, they produce an infinitely many solutions defined by:
[tex]\[ y = -x - 3 \][/tex]
Any point [tex]\((x, -x - 3)\)[/tex] is a solution to these equations.
#### Pair 3: [tex]\( -x - y = 3 \)[/tex] and [tex]\( x + y = 7 \)[/tex]
Let's re-express and compare these equations:
[tex]\[ -x - y = 3 \][/tex]
[tex]\[ x + y = 7 \][/tex]
If we add these equations together:
[tex]\[ (-x - y) + (x + y) = 3 + 7 \implies 0 = 10 \][/tex]
This is another contradiction, which means there is no solution for this pair of equations.
### Summary of Solutions:
1. The pair [tex]\( x + y = -3 \)[/tex] and [tex]\( x + y = 4 \)[/tex] leads to no solution (contradiction).
2. The pair [tex]\( x + y = -3 \)[/tex] and [tex]\( -x - y = 3 \)[/tex] has infinitely many solutions with [tex]\( y = -x - 3 \)[/tex].
3. The pair [tex]\( -x - y = 3 \)[/tex] and [tex]\( x + y = 7 \)[/tex] leads to no solution (contradiction).
So, the summarized result for these systems of equations is:
- One solution: [tex]\( [] \)[/tex]
- No solution: [tex]\( [] \)[/tex]
- Infinitely many solutions defined by [tex]\( y = -x - 3 \)[/tex]
This analysis explains why the results are: no solution for pairs 1 and 3, and infinitely many solutions for pair 2.
