At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Get quick and reliable solutions to your questions from a community of experienced experts on our platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To determine the solutions to the system of equations and sort the given coordinate points based on whether they are solutions or not, we follow these steps:
First, we need to solve the following system of equations:
[tex]\[ \begin{cases} 2x + y = 10 \quad \text{(Equation 1)} \\ 3x - y = 5 \quad \text{(Equation 2)} \end{cases} \][/tex]
1. Solving the system of equations:
Start by solving Equation 1 for [tex]\( y \)[/tex]:
[tex]\[ y = 10 - 2x \][/tex]
Substitute [tex]\( y = 10 - 2x \)[/tex] into Equation 2:
[tex]\[ 3x - (10 - 2x) = 5 \][/tex]
Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 3x - 10 + 2x = 5 \\ 5x - 10 = 5 \\ 5x = 15 \\ x = 3 \][/tex]
Substitute [tex]\( x = 3 \)[/tex] back into Equation 1 to find [tex]\( y \)[/tex]:
[tex]\[ 2(3) + y = 10 \\ 6 + y = 10 \\ y = 4 \][/tex]
Therefore, the solution to the system is [tex]\( (x, y) = (3, 4) \)[/tex].
2. Identifying the solution and non-solution points:
Let's check each given point to see if it satisfies both equations.
- [tex]\( (14, 1) \)[/tex]
[tex]\[ 2(14) + 1 = 28 + 1 = 29 \quad \text{(not 10)} \\ 3(14) - 1 = 42 - 1 = 41 \quad \text{(not 5)} \][/tex]
Hence, [tex]\( (14, 1) \)[/tex] is not a solution.
- [tex]\( (3, 7) \)[/tex]
[tex]\[ 2(3) + 7 = 6 + 7 = 13 \quad \text{(not 10)} \\ 3(3) - 7 = 9 - 7 = 2 \quad \text{(not 5)} \][/tex]
Hence, [tex]\( (3, 7) \)[/tex] is not a solution.
- [tex]\( (1, 3) \)[/tex]
[tex]\[ 2(1) + 3 = 2 + 3 = 5 \quad \text{(not 10)} \\ 3(1) - 3 = 3 - 3 = 0 \quad \text{(not 5)} \][/tex]
Hence, [tex]\( (1, 3) \)[/tex] is not a solution.
- [tex]\( (2, 9) \)[/tex]
[tex]\[ 2(2) + 9 = 4 + 9 = 13 \quad \text{(not 10)} \\ 3(2) - 9 = 6 - 9 = -3 \quad \text{(not 5)} \][/tex]
Hence, [tex]\( (2, 9) \)[/tex] is not a solution.
- [tex]\( (5, 4) \)[/tex]
[tex]\[ 2(5) + 4 = 10 + 4 = 14 \quad \text{(not 10)} \\ 3(5) - 4 = 15 - 4 = 11 \quad \text{(not 5)} \][/tex]
Hence, [tex]\( (5, 4) \)[/tex] is not a solution.
- [tex]\( (1, 13) \)[/tex]
[tex]\[ 2(1) + 13 = 2 + 13 = 15 \quad \text{(not 10)} \\ 3(1) - 13 = 3 - 13 = -10 \quad \text{(not 5)} \][/tex]
Hence, [tex]\( (1, 13) \)[/tex] is not a solution.
- [tex]\( (2, 4) \)[/tex]
[tex]\[ 2(2) + 4 = 4 + 4 = 8 \quad \text{(not 10)} \\ 3(2) - 4 = 6 - 4 = 2 \quad \text{(not 5)} \][/tex]
Hence, [tex]\( (2, 4) \)[/tex] is not a solution.
- [tex]\( (7, 7) \)[/tex]
[tex]\[ 2(7) + 7 = 14 + 7 = 21 \quad \text{(not 10)} \\ 3(7) - 7 = 21 - 7 = 14 \quad \text{(not 5)} \][/tex]
Hence, [tex]\( (7, 7) \)[/tex] is not a solution.
3. Summary:
- Solution to the System:
[tex]\[ (3, 4) \][/tex]
- Not a Solution to the System:
[tex]\[ (14, 1), (3, 7), (1, 3), (2, 9), (5, 4), (1, 13), (2, 4), (7, 7) \][/tex]
First, we need to solve the following system of equations:
[tex]\[ \begin{cases} 2x + y = 10 \quad \text{(Equation 1)} \\ 3x - y = 5 \quad \text{(Equation 2)} \end{cases} \][/tex]
1. Solving the system of equations:
Start by solving Equation 1 for [tex]\( y \)[/tex]:
[tex]\[ y = 10 - 2x \][/tex]
Substitute [tex]\( y = 10 - 2x \)[/tex] into Equation 2:
[tex]\[ 3x - (10 - 2x) = 5 \][/tex]
Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 3x - 10 + 2x = 5 \\ 5x - 10 = 5 \\ 5x = 15 \\ x = 3 \][/tex]
Substitute [tex]\( x = 3 \)[/tex] back into Equation 1 to find [tex]\( y \)[/tex]:
[tex]\[ 2(3) + y = 10 \\ 6 + y = 10 \\ y = 4 \][/tex]
Therefore, the solution to the system is [tex]\( (x, y) = (3, 4) \)[/tex].
2. Identifying the solution and non-solution points:
Let's check each given point to see if it satisfies both equations.
- [tex]\( (14, 1) \)[/tex]
[tex]\[ 2(14) + 1 = 28 + 1 = 29 \quad \text{(not 10)} \\ 3(14) - 1 = 42 - 1 = 41 \quad \text{(not 5)} \][/tex]
Hence, [tex]\( (14, 1) \)[/tex] is not a solution.
- [tex]\( (3, 7) \)[/tex]
[tex]\[ 2(3) + 7 = 6 + 7 = 13 \quad \text{(not 10)} \\ 3(3) - 7 = 9 - 7 = 2 \quad \text{(not 5)} \][/tex]
Hence, [tex]\( (3, 7) \)[/tex] is not a solution.
- [tex]\( (1, 3) \)[/tex]
[tex]\[ 2(1) + 3 = 2 + 3 = 5 \quad \text{(not 10)} \\ 3(1) - 3 = 3 - 3 = 0 \quad \text{(not 5)} \][/tex]
Hence, [tex]\( (1, 3) \)[/tex] is not a solution.
- [tex]\( (2, 9) \)[/tex]
[tex]\[ 2(2) + 9 = 4 + 9 = 13 \quad \text{(not 10)} \\ 3(2) - 9 = 6 - 9 = -3 \quad \text{(not 5)} \][/tex]
Hence, [tex]\( (2, 9) \)[/tex] is not a solution.
- [tex]\( (5, 4) \)[/tex]
[tex]\[ 2(5) + 4 = 10 + 4 = 14 \quad \text{(not 10)} \\ 3(5) - 4 = 15 - 4 = 11 \quad \text{(not 5)} \][/tex]
Hence, [tex]\( (5, 4) \)[/tex] is not a solution.
- [tex]\( (1, 13) \)[/tex]
[tex]\[ 2(1) + 13 = 2 + 13 = 15 \quad \text{(not 10)} \\ 3(1) - 13 = 3 - 13 = -10 \quad \text{(not 5)} \][/tex]
Hence, [tex]\( (1, 13) \)[/tex] is not a solution.
- [tex]\( (2, 4) \)[/tex]
[tex]\[ 2(2) + 4 = 4 + 4 = 8 \quad \text{(not 10)} \\ 3(2) - 4 = 6 - 4 = 2 \quad \text{(not 5)} \][/tex]
Hence, [tex]\( (2, 4) \)[/tex] is not a solution.
- [tex]\( (7, 7) \)[/tex]
[tex]\[ 2(7) + 7 = 14 + 7 = 21 \quad \text{(not 10)} \\ 3(7) - 7 = 21 - 7 = 14 \quad \text{(not 5)} \][/tex]
Hence, [tex]\( (7, 7) \)[/tex] is not a solution.
3. Summary:
- Solution to the System:
[tex]\[ (3, 4) \][/tex]
- Not a Solution to the System:
[tex]\[ (14, 1), (3, 7), (1, 3), (2, 9), (5, 4), (1, 13), (2, 4), (7, 7) \][/tex]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
