Determine the number of solutions of this system of linear equations:
−x+y=3
−2x+2y=3
This question requires you to show your work.
(1 point)
Responses
Infinitely Many Solutions
Infinitely Many Solutions
No Solutions
No Solutions
One Solution
To determine the number of solutions, we can rewrite the system of equations in matrix form:
| -1 1 | | x | = | 3 |
| -2 2 | | y | | 3 |
We can then find the determinant of the coefficient matrix:
det| -1 1 |
| -2 2 |
= (-1)(2) - (1)(-2)
= -2 + 2
= 0
Since the determinant is 0, the system of equations has either no solutions or infinitely many solutions. To determine which it is, we can check the consistency of the system.
If we multiply the first equation by -2, we get:
2x - 2y = -6
Comparing this with the second equation:
-2x + 2y = 3
We can see that they are equivalent. This means the system is indeed consistent and has infinitely many solutions.
Therefore, the system of linear equations has Infinitely Many Solutions.
explain
In this system of linear equations:
Equation 1: -x + y = 3
Equation 2: -2x + 2y = 3
We can see that Equation 2 is simply Equation 1 multiplied by 2. This means that both equations represent the same line in the xy-plane. Since both equations are representing the same line, they have an infinite number of solutions because every point on the line is a solution to the system.
This is why the system has Infinitely Many Solutions.
Determine the number of solutions of this system of linear equations:
−x+y=3
−2x+2y=3
This question requires you to show your work.
Responses
Infinitely Many Solutions
Infinitely Many Solutions
No Solutions
No Solutions
One Solution
To determine the number of solutions, we can rewrite the system of equations as a matrix:
| -1 1 | | x | = | 3 |
| -2 2 | | y | | 3 |
We can calculate the determinant of the coefficient matrix:
det| -1 1 |
| -2 2 |
= (-1)(2) - (1)(-2)
= -2 - (-2)
= 0
Since the determinant is equal to 0, the system may have either no solutions or infinitely many solutions. To determine which of these possibilities occur, we need to check the consistency of the system of equations.
Let's rewrite Equation 1:
-x + y = 3
y = x + 3
Substitute y = x + 3 into Equation 2:
-2x + 2(x + 3) = 3
-2x + 2x + 6 = 3
6 = 3
This leads to a contradiction (6 is not equal to 3), which means there are no solutions to the system of equations.
Therefore, the correct answer is No Solutions.