Plug a few points into the original equations and check to show that the system of
equations has one solution. Explain your work.
To demonstrate that the system of equations has one solution, we need to select a few points and substitute them into the original equations to see if they satisfy both equations simultaneously.
Let's assume we have the following system of equations:
Equation 1: 2x + y = 5
Equation 2: x - 3y = -2
Point 1: (2, 1)
Substituting these values into the equations:
Equation 1: 2(2) + 1 = 5
4 + 1 = 5
5 = 5 (True)
Equation 2: 2 - 3(1) = -2
2 - 3 = -2
-1 = -2 (False)
As Equation 2 does not hold true for our selected point, (2, 1) is not a solution for the system of equations.
Point 2: (3, -1)
Substituting these values into the equations:
Equation 1: 2(3) + (-1) = 5
6 - 1 = 5
5 = 5 (True)
Equation 2: 3 - 3(-1) = -2
3 + 3 = -2
6 = -2 (False)
As Equation 2 does not hold true for our selected point, (3, -1) is not a solution for the system of equations.
Point 3: (-1, 3)
Substituting these values into the equations:
Equation 1: 2(-1) + 3 = 5
-2 + 3 = 5
1 = 5 (False)
Equation 2: -1 - 3(3) = -2
-1 - 9 = -2
-10 = -2 (False)
As neither Equation 1 nor Equation 2 hold true for our selected point, (-1, 3) is not a solution for the system of equations.
Based on these test results, we see that none of the selected points satisfy both equations simultaneously. This indicates that the system of equations does not have a unique solution.
To check if a system of equations has one solution, we need to plug a few points into the original equations and determine if they satisfy all the equations simultaneously.
Let's assume we have a system of equations with two variables, x and y, given by:
Equation 1: 2x - 3y = 4
Equation 2: 5x + y = -1
To show that this system has one solution, we need to find a specific point (x, y) that satisfies both equations.
Let's choose the point (1, -6) and substitute these values into the equations:
For Equation 1:
2(1) - 3(-6) = 2 + 18 = 20
For Equation 2:
5(1) + (-6) = 5 - 6 = -1
Since both equations are satisfied, meaning both sides of each equation are equal, we can conclude that the point (1, -6) is a solution to the system of equations.
This verifies that the system has one solution.