Which linear system has the solution [tex][tex]$x=5$[/tex][/tex] and [tex][tex]$y=-4$[/tex][/tex]?

A.
[tex]x + 3y = 12[/tex]
[tex]4x - 2y = -27[/tex]

B.
[tex]2x + 3y = 5[/tex]
[tex]2x + 4y = 10[/tex]

C.
[tex]x + 3y = 5[/tex]
[tex]-2x + y = 11[/tex]

D.
[tex]3x + y = 11[/tex]
[tex]-2x + 4y = -26[/tex]



Answer :

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]