To determine which linear system has the solution [tex]\( x = 5 \)[/tex] and [tex]\( y = -4 \)[/tex], we need to substitute these values into each of the given systems and check if the equations are satisfied.
System a:
1. [tex]\( x + 3y = 12 \)[/tex]
2. [tex]\( 4x - 2y = -27 \)[/tex]
Substituting [tex]\( x = 5 \)[/tex] and [tex]\( y = -4 \)[/tex]:
1. [tex]\( 5 + 3(-4) \)[/tex]
[tex]\( = 5 - 12 \)[/tex]
[tex]\( = -7 \)[/tex]
This does not equal [tex]\(12\)[/tex], so system [tex]\(a\)[/tex] is not satisfied.
System b:
1. [tex]\( 2x + 3y = 5 \)[/tex]
2. [tex]\( 2x + 4y = 10 \)[/tex]
3. [tex]\( -2x + y = 11 \)[/tex]
Substituting [tex]\( x = 5 \)[/tex] and [tex]\( y = -4 \)[/tex]:
1. [tex]\( 2(5) + 3(-4) \)[/tex]
[tex]\( = 10 - 12 \)[/tex]
[tex]\( = -2 \)[/tex]
This does not equal [tex]\(5\)[/tex], so system [tex]\(b\)[/tex] is not satisfied.
System c:
1. [tex]\( x + 3y = 5 \)[/tex]
Substituting [tex]\( x = 5 \)[/tex] and [tex]\( y = -4 \)[/tex]:
1. [tex]\( 5 + 3(-4) \)[/tex]
[tex]\( = 5 - 12 \)[/tex]
[tex]\( = -7 \)[/tex]
This does not equal [tex]\(5\)[/tex], so system [tex]\(c\)[/tex] is not satisfied.
System d:
1. [tex]\( 3x + y = 11 \)[/tex]
2. [tex]\( -2x + 4y = -26 \)[/tex]
Substituting [tex]\( x = 5 \)[/tex] and [tex]\( y = -4 \)[/tex]:
1. [tex]\( 3(5) + (-4) \)[/tex]
[tex]\( = 15 - 4 \)[/tex]
[tex]\( = 11 \)[/tex] (matches the first equation)
2. [tex]\( -2(5) + 4(-4) \)[/tex]
[tex]\( = -10 - 16 \)[/tex]
[tex]\( = -26 \)[/tex] (matches the second equation)
Both equations are satisfied, so system [tex]\(d\)[/tex] is satisfied.
Therefore, the linear system that has the solution [tex]\( x = 5 \)[/tex] and [tex]\( y = -4 \)[/tex] is system (d):
[tex]\[
3x + y = 11
\][/tex]
[tex]\[
-2x + 4y = -26
\][/tex]
