To determine which linear system has the solution [tex]\( x = 5 \)[/tex] and [tex]\( y = -4 \)[/tex], we need to verify each system by substituting these values into the equations and checking if they satisfy all equations in the system.
### System a:
1. [tex]\( x + 3y = 12 \)[/tex]
2. [tex]\( 4x - 2y = -27 \)[/tex]
Substitute [tex]\( x = 5 \)[/tex] and [tex]\( y = -4 \)[/tex] into these equations:
1. [tex]\( 5 + 3(-4) = 5 - 12 = -7 \neq 12 \)[/tex]
2. [tex]\( 4(5) - 2(-4) = 20 + 8 = 28 \neq -27 \)[/tex]
System a is not satisfied.
### System b:
1. [tex]\( 2x + 3y = 5 \)[/tex]
2. [tex]\( 2x + 4y = 10 \)[/tex]
3. [tex]\( -2x + y = 11 \)[/tex]
Substitute [tex]\( x = 5 \)[/tex] and [tex]\( y = -4 \)[/tex] into these equations:
1. [tex]\( 2(5) + 3(-4) = 10 - 12 = -2 \neq 5 \)[/tex]
2. [tex]\( 2(5) + 4(-4) = 10 - 16 = -6 \neq 10 \)[/tex]
3. [tex]\( -2(5) + (-4) = -10 - 4 = -14 \neq 11 \)[/tex]
System b is not satisfied.
### System c:
1. [tex]\( x + 3y = 5 \)[/tex]
2. [tex]\( 2x + 4y = 10 \)[/tex]
3. [tex]\( -2x + y = 11 \)[/tex]
Substitute [tex]\( x = 5 \)[/tex] and [tex]\( y = -4 \)[/tex] into these equations:
1. [tex]\( 5 + 3(-4) = 5 - 12 = -7 \neq 5 \)[/tex]
2. [tex]\( 2(5) + 4(-4) = 10 - 16 = -6 \neq 10 \)[/tex]
3. [tex]\( -2(5) + (-4) = -10 - 4 = -14 \neq 11 \)[/tex]
System c is not satisfied.
### System d:
1. [tex]\( 3x + y = 11 \)[/tex]
2. [tex]\( -2x + 4y = -26 \)[/tex]
Substitute [tex]\( x = 5 \)[/tex] and [tex]\( y = -4 \)[/tex] into these equations:
1. [tex]\( 3(5) + (-4) = 15 - 4 = 11 \)[/tex]
2. [tex]\( -2(5) + 4(-4) = -10 - 16 = -26 \)[/tex]
Both equations are satisfied with the solution [tex]\( x = 5 \)[/tex] and [tex]\( y = -4 \)[/tex].
Thus, the linear system that has the solution [tex]\( x = 5 \)[/tex] and [tex]\( y = -4 \)[/tex] is system [tex]\((d)\)[/tex]:
[tex]\[ 3x + y = 11 \][/tex]
[tex]\[ -2x + 4y = -26 \][/tex]
