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Sagot :
To solve the systems of equations, let's address each pair individually and find the solutions step-by-step:
### Pair 1:
[tex]\[ \begin{aligned} 2x + y &= 12 \\ x &= 9 - 2y \end{aligned} \][/tex]
Substitute [tex]\( x = 9 - 2y \)[/tex] into [tex]\( 2x + y = 12 \)[/tex]:
[tex]\[ 2(9 - 2y) + y = 12 \\ 18 - 4y + y = 12 \\ 18 - 3y = 12 \\ -3y = 12 - 18 \\ -3y = -6 \\ y = 2 \][/tex]
Using [tex]\( y = 2 \)[/tex] in [tex]\( x = 9 - 2y \)[/tex]:
[tex]\[ x = 9 - 2(2) \\ x = 9 - 4 \\ x = 5 \][/tex]
Solution: [tex]\( (x, y) = (5, 2) \)[/tex].
### Pair 2:
[tex]\[ \begin{aligned} 2x + y &= 11 \\ x - 2y &= -7 \end{aligned} \][/tex]
Multiply the second equation by 2 to eliminate [tex]\( y \)[/tex]:
[tex]\[ 2(x - 2y) = 2(-7) \\ 2x - 4y = -14 \][/tex]
Subtract the second equation from the first:
[tex]\[ (2x + y) - (2x - 4y) = 11 - (-14) \\ 2x + y - 2x + 4y = 11 + 14 \\ 5y = 25 \\ y = 5 \][/tex]
Using [tex]\( y = 5 \)[/tex] in [tex]\( x - 2y = -7 \)[/tex]:
[tex]\[ x - 2(5) = -7 \\ x - 10 = -7 \\ x = 3 \][/tex]
Solution: [tex]\( (x, y) = (3, 5) \)[/tex].
### Pair 3:
[tex]\[ \begin{aligned} y &= 11 - 2x \\ 4x - 3y &= -13 \end{aligned} \][/tex]
Substitute [tex]\( y = 11 - 2x \)[/tex] into [tex]\( 4x - 3y = -13 \)[/tex]:
[tex]\[ 4x - 3(11 - 2x) = -13 \\ 4x - 33 + 6x = -13 \\ 10x - 33 = -13 \\ 10x = 20 \\ x = 2 \][/tex]
Using [tex]\( x = 2 \)[/tex] in [tex]\( y = 11 - 2x \)[/tex]:
[tex]\[ y = 11 - 2(2) \\ y = 11 - 4 \\ y = 7 \][/tex]
Solution: [tex]\( (x, y) = (2, 7) \)[/tex].
### Pair 4:
[tex]\[ \begin{aligned} x + 3y &= 16 \\ 2x - y &= 11 \end{aligned} \][/tex]
Multiply the second equation by 3 to eliminate [tex]\( y \)[/tex]:
[tex]\[ 3(2x - y) = 3(11) \\ 6x - 3y = 33 \][/tex]
Add the first equation to the modified second equation:
[tex]\[ (x + 3y) + (6x - 3y) = 16 + 33 \\ 7x = 49 \\ x = 7 \][/tex]
Using [tex]\( x = 7 \)[/tex] in [tex]\( 2x - y = 11 \)[/tex]:
[tex]\[ 2(7) - y = 11 \\ 14 - y = 11 \\ y = 3 \][/tex]
Solution: [tex]\( (x, y) = (7, 3) \)[/tex].
### Summary of Matches:
- [tex]\(\begin{aligned} 2x + y &= 12 \\ x &= 9 - 2y \end{aligned}\)[/tex] : [tex]\( (x, y) = (5, 2) \)[/tex]
- [tex]\(\begin{aligned} 2x + y &= 11 \\ x - 2y &= -7 \end{aligned}\)[/tex] : [tex]\( (x, y) = (3, 5) \)[/tex]
- [tex]\(\begin{aligned} y &= 11 - 2x \\ 4x - 3y &= -13 \end{aligned} \)[/tex] : [tex]\( (x, y) = (2, 7) \)[/tex]
- [tex]\(\begin{aligned} x + 3y &= 16 \\ 2x - y &= 11 \end{aligned} \)[/tex] : [tex]\( (x, y) = (7, 3) \)[/tex]
Thus, drag the tiles accordingly:
- [tex]\( 2x + y = 12, x = 9 - 2y \)[/tex] : [tex]\( (x, y) = (5, 2) \)[/tex]
- [tex]\( 2x + y = 11, x - 2y = -7 \)[/tex] : [tex]\( (x, y) = (3, 5) \)[/tex]
- [tex]\( y = 11 - 2x, 4x - 3y = -13 \)[/tex] : [tex]\( (x, y) = (2, 7) \)[/tex]
- [tex]\( x + 3y = 16, 2x - y = 11 \)[/tex] : [tex]\( (x, y) = (7, 3) \)[/tex]
### Pair 1:
[tex]\[ \begin{aligned} 2x + y &= 12 \\ x &= 9 - 2y \end{aligned} \][/tex]
Substitute [tex]\( x = 9 - 2y \)[/tex] into [tex]\( 2x + y = 12 \)[/tex]:
[tex]\[ 2(9 - 2y) + y = 12 \\ 18 - 4y + y = 12 \\ 18 - 3y = 12 \\ -3y = 12 - 18 \\ -3y = -6 \\ y = 2 \][/tex]
Using [tex]\( y = 2 \)[/tex] in [tex]\( x = 9 - 2y \)[/tex]:
[tex]\[ x = 9 - 2(2) \\ x = 9 - 4 \\ x = 5 \][/tex]
Solution: [tex]\( (x, y) = (5, 2) \)[/tex].
### Pair 2:
[tex]\[ \begin{aligned} 2x + y &= 11 \\ x - 2y &= -7 \end{aligned} \][/tex]
Multiply the second equation by 2 to eliminate [tex]\( y \)[/tex]:
[tex]\[ 2(x - 2y) = 2(-7) \\ 2x - 4y = -14 \][/tex]
Subtract the second equation from the first:
[tex]\[ (2x + y) - (2x - 4y) = 11 - (-14) \\ 2x + y - 2x + 4y = 11 + 14 \\ 5y = 25 \\ y = 5 \][/tex]
Using [tex]\( y = 5 \)[/tex] in [tex]\( x - 2y = -7 \)[/tex]:
[tex]\[ x - 2(5) = -7 \\ x - 10 = -7 \\ x = 3 \][/tex]
Solution: [tex]\( (x, y) = (3, 5) \)[/tex].
### Pair 3:
[tex]\[ \begin{aligned} y &= 11 - 2x \\ 4x - 3y &= -13 \end{aligned} \][/tex]
Substitute [tex]\( y = 11 - 2x \)[/tex] into [tex]\( 4x - 3y = -13 \)[/tex]:
[tex]\[ 4x - 3(11 - 2x) = -13 \\ 4x - 33 + 6x = -13 \\ 10x - 33 = -13 \\ 10x = 20 \\ x = 2 \][/tex]
Using [tex]\( x = 2 \)[/tex] in [tex]\( y = 11 - 2x \)[/tex]:
[tex]\[ y = 11 - 2(2) \\ y = 11 - 4 \\ y = 7 \][/tex]
Solution: [tex]\( (x, y) = (2, 7) \)[/tex].
### Pair 4:
[tex]\[ \begin{aligned} x + 3y &= 16 \\ 2x - y &= 11 \end{aligned} \][/tex]
Multiply the second equation by 3 to eliminate [tex]\( y \)[/tex]:
[tex]\[ 3(2x - y) = 3(11) \\ 6x - 3y = 33 \][/tex]
Add the first equation to the modified second equation:
[tex]\[ (x + 3y) + (6x - 3y) = 16 + 33 \\ 7x = 49 \\ x = 7 \][/tex]
Using [tex]\( x = 7 \)[/tex] in [tex]\( 2x - y = 11 \)[/tex]:
[tex]\[ 2(7) - y = 11 \\ 14 - y = 11 \\ y = 3 \][/tex]
Solution: [tex]\( (x, y) = (7, 3) \)[/tex].
### Summary of Matches:
- [tex]\(\begin{aligned} 2x + y &= 12 \\ x &= 9 - 2y \end{aligned}\)[/tex] : [tex]\( (x, y) = (5, 2) \)[/tex]
- [tex]\(\begin{aligned} 2x + y &= 11 \\ x - 2y &= -7 \end{aligned}\)[/tex] : [tex]\( (x, y) = (3, 5) \)[/tex]
- [tex]\(\begin{aligned} y &= 11 - 2x \\ 4x - 3y &= -13 \end{aligned} \)[/tex] : [tex]\( (x, y) = (2, 7) \)[/tex]
- [tex]\(\begin{aligned} x + 3y &= 16 \\ 2x - y &= 11 \end{aligned} \)[/tex] : [tex]\( (x, y) = (7, 3) \)[/tex]
Thus, drag the tiles accordingly:
- [tex]\( 2x + y = 12, x = 9 - 2y \)[/tex] : [tex]\( (x, y) = (5, 2) \)[/tex]
- [tex]\( 2x + y = 11, x - 2y = -7 \)[/tex] : [tex]\( (x, y) = (3, 5) \)[/tex]
- [tex]\( y = 11 - 2x, 4x - 3y = -13 \)[/tex] : [tex]\( (x, y) = (2, 7) \)[/tex]
- [tex]\( x + 3y = 16, 2x - y = 11 \)[/tex] : [tex]\( (x, y) = (7, 3) \)[/tex]
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