Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Discover the answers you need from a community of experts ready to help you with their knowledge and experience in various fields. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
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]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
