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]
