Answer :
To determine the correct value of [tex]\( k \)[/tex], we need to solve the system of equations given by:
[tex]\[ \begin{cases} (k+1)x + 2y = 7 \\ x - 4y = -7 \end{cases} \][/tex]
We are given that [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex].
First, let's substitute these values into the second equation to confirm the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x - 4y = -7 \][/tex]
Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex]:
[tex]\[ 2 - 4(1) = -7 \][/tex]
[tex]\[ 2 - 4 = -2 \neq -7 \][/tex]
There seems to be an inconsistency here considering the standard approach. However, let's proceed with the first equation to find the value of [tex]\( k \)[/tex] given the equations match the constants provided.
Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex] into the first equation:
[tex]\[ (k+1)x + 2y = 7 \][/tex]
Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex]:
[tex]\[ (k+1)(2) + 2(1) = 7 \][/tex]
Simplify:
[tex]\[ 2(k+1) + 2 = 7 \][/tex]
[tex]\[ 2k + 2 + 2 = 7 \][/tex]
[tex]\[ 2k + 4 = 7 \][/tex]
Subtract 4 from both sides:
[tex]\[ 2k = 3 \][/tex]
Divide by 2:
[tex]\[ k = \frac{3}{2} \][/tex]
Thus, there is no integer value of [tex]\( k \)[/tex] from the given options ([tex]\( k = 0 \)[/tex], [tex]\( k = 1 \)[/tex], [tex]\( k = 2 \)[/tex], [tex]\( k = 3 \)[/tex]) that satisfies these equations with [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex]. Hence, the correct option for [tex]\( k \)[/tex] does not exist within the provided choices.
But since we are to choose from the given options, we conclude that:
None of the given values of [tex]\( k \)[/tex] satisfy the equations with [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex].
Thus the correct answer is:
```
[]
```
[tex]\[ \begin{cases} (k+1)x + 2y = 7 \\ x - 4y = -7 \end{cases} \][/tex]
We are given that [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex].
First, let's substitute these values into the second equation to confirm the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x - 4y = -7 \][/tex]
Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex]:
[tex]\[ 2 - 4(1) = -7 \][/tex]
[tex]\[ 2 - 4 = -2 \neq -7 \][/tex]
There seems to be an inconsistency here considering the standard approach. However, let's proceed with the first equation to find the value of [tex]\( k \)[/tex] given the equations match the constants provided.
Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex] into the first equation:
[tex]\[ (k+1)x + 2y = 7 \][/tex]
Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex]:
[tex]\[ (k+1)(2) + 2(1) = 7 \][/tex]
Simplify:
[tex]\[ 2(k+1) + 2 = 7 \][/tex]
[tex]\[ 2k + 2 + 2 = 7 \][/tex]
[tex]\[ 2k + 4 = 7 \][/tex]
Subtract 4 from both sides:
[tex]\[ 2k = 3 \][/tex]
Divide by 2:
[tex]\[ k = \frac{3}{2} \][/tex]
Thus, there is no integer value of [tex]\( k \)[/tex] from the given options ([tex]\( k = 0 \)[/tex], [tex]\( k = 1 \)[/tex], [tex]\( k = 2 \)[/tex], [tex]\( k = 3 \)[/tex]) that satisfies these equations with [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex]. Hence, the correct option for [tex]\( k \)[/tex] does not exist within the provided choices.
But since we are to choose from the given options, we conclude that:
None of the given values of [tex]\( k \)[/tex] satisfy the equations with [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex].
Thus the correct answer is:
```
[]
```
