Answer :
Alright, let's analyze each of the given systems of linear equations step-by-step and determine the number of solutions for each system.
### System 1:
[tex]\[ y = -2x + 5 \][/tex]
[tex]\[ 2x + y = -7 \][/tex]
Step-by-Step:
1. Substitute [tex]\( y = -2x + 5 \)[/tex] into [tex]\( 2x + y = -7 \)[/tex]:
[tex]\[ 2x + (-2x + 5) = -7 \][/tex]
2. Simplify the equation:
[tex]\[ 5 = -7 \][/tex]
This results in a contradiction. Therefore, the first system has no solution.
### System 2:
[tex]\[ y = x + 6 \][/tex]
[tex]\[ 3x - 3y = -18 \][/tex]
Step-by-Step:
1. Substitute [tex]\( y = x + 6 \)[/tex] into [tex]\( 3x - 3y = -18 \)[/tex]:
[tex]\[ 3x - 3(x + 6) = -18 \][/tex]
2. Simplify the equation:
[tex]\[ 3x - 3x - 18 = -18 \implies -18 = -18 \][/tex]
This is an identity, which means the equations are dependent, and every solution of the first equation is also a solution of the second equation. Thus, the second system has infinitely many solutions.
### System 3:
[tex]\[ y = -4x + 11 \][/tex]
[tex]\[ -6x + y = 11 \][/tex]
Step-by-Step:
1. Substitute [tex]\( y = -4x + 11 \)[/tex] into [tex]\( -6x + y = 11 \)[/tex]:
[tex]\[ -6x + (-4x + 11) = 11 \][/tex]
2. Simplify the equation:
[tex]\[ -6x - 4x + 11 = 11 \implies -10x + 11 = 11 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[ -10x = 0 \implies x = 0 \][/tex]
4. Substitute [tex]\( x = 0 \)[/tex] back into [tex]\( y = -4x + 11 \)[/tex]:
[tex]\[ y = -4(0) + 11 = 11 \][/tex]
So, the solution is [tex]\( (0, 11) \)[/tex]. Therefore, the third system has one solution.
### Summary
- The system [tex]\( y = -2x + 5 \)[/tex] and [tex]\( 2x + y = -7 \)[/tex] has no solution.
- The system [tex]\( y = x + 6 \)[/tex] and [tex]\( 3x - 3y = -18 \)[/tex] has infinitely many solutions.
- The system [tex]\( y = -4x + 11 \)[/tex] and [tex]\( -6x + y = 11 \)[/tex] has one solution.
Thus, the systems match with the number of solutions as follows:
- [tex]\( y = -2x + 5 \)[/tex] and [tex]\( 2x + y = -7 \)[/tex] → no solution
- [tex]\( y = x + 6 \)[/tex] and [tex]\( 3x - 3y = -18 \)[/tex] → infinitely many solutions
- [tex]\( y = -4x + 11 \)[/tex] and [tex]\( -6x + y = 11 \)[/tex] → one solution
### System 1:
[tex]\[ y = -2x + 5 \][/tex]
[tex]\[ 2x + y = -7 \][/tex]
Step-by-Step:
1. Substitute [tex]\( y = -2x + 5 \)[/tex] into [tex]\( 2x + y = -7 \)[/tex]:
[tex]\[ 2x + (-2x + 5) = -7 \][/tex]
2. Simplify the equation:
[tex]\[ 5 = -7 \][/tex]
This results in a contradiction. Therefore, the first system has no solution.
### System 2:
[tex]\[ y = x + 6 \][/tex]
[tex]\[ 3x - 3y = -18 \][/tex]
Step-by-Step:
1. Substitute [tex]\( y = x + 6 \)[/tex] into [tex]\( 3x - 3y = -18 \)[/tex]:
[tex]\[ 3x - 3(x + 6) = -18 \][/tex]
2. Simplify the equation:
[tex]\[ 3x - 3x - 18 = -18 \implies -18 = -18 \][/tex]
This is an identity, which means the equations are dependent, and every solution of the first equation is also a solution of the second equation. Thus, the second system has infinitely many solutions.
### System 3:
[tex]\[ y = -4x + 11 \][/tex]
[tex]\[ -6x + y = 11 \][/tex]
Step-by-Step:
1. Substitute [tex]\( y = -4x + 11 \)[/tex] into [tex]\( -6x + y = 11 \)[/tex]:
[tex]\[ -6x + (-4x + 11) = 11 \][/tex]
2. Simplify the equation:
[tex]\[ -6x - 4x + 11 = 11 \implies -10x + 11 = 11 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[ -10x = 0 \implies x = 0 \][/tex]
4. Substitute [tex]\( x = 0 \)[/tex] back into [tex]\( y = -4x + 11 \)[/tex]:
[tex]\[ y = -4(0) + 11 = 11 \][/tex]
So, the solution is [tex]\( (0, 11) \)[/tex]. Therefore, the third system has one solution.
### Summary
- The system [tex]\( y = -2x + 5 \)[/tex] and [tex]\( 2x + y = -7 \)[/tex] has no solution.
- The system [tex]\( y = x + 6 \)[/tex] and [tex]\( 3x - 3y = -18 \)[/tex] has infinitely many solutions.
- The system [tex]\( y = -4x + 11 \)[/tex] and [tex]\( -6x + y = 11 \)[/tex] has one solution.
Thus, the systems match with the number of solutions as follows:
- [tex]\( y = -2x + 5 \)[/tex] and [tex]\( 2x + y = -7 \)[/tex] → no solution
- [tex]\( y = x + 6 \)[/tex] and [tex]\( 3x - 3y = -18 \)[/tex] → infinitely many solutions
- [tex]\( y = -4x + 11 \)[/tex] and [tex]\( -6x + y = 11 \)[/tex] → one solution
