Hi! Can someone check these two answers? Thanks!
Directions: Find the correct number of solutions for the following system of equations.
1.) 5x - 3y + 2z = 3
2x + 4y-z = 7
x - 11y + 4z = 3
2.) 2x + y- 3z = 4
4x + 2z = 10
-2x + 3y - 13z = -8
A.) One Solution
B.) Infinitely Many Solutions
C.) No Solution
My answers:
1.) C.
2.) A.
#1 C
#2 B
4x + 2z = 10, so z = 5-2x
That gives
2x + y - 3(5-2x) = 4
-2x + 3y - 13(5-2x) = -8
or
8x + y = 19
24x + 3y = 57
so, y = 19-8x
That is, whatever x you choose, y and z can be found.
To check the answers for the given system of equations, you can use determinant to determine the number of solutions.
For the first system of equations:
1.) 5x - 3y + 2z = 3
2x + 4y - z = 7
x - 11y + 4z = 3
To find the determinant, arrange the coefficients of the variables in a 3x3 matrix:
| 5 -3 2 |
| 2 4 -1 |
| 1 -11 4 |
Using any method (cofactor expansion, row reduction, etc.), find the determinant of this matrix. If the determinant is nonzero, the system has one unique solution. If the determinant is zero, the system has either infinitely many solutions or no solutions.
For this system, using cofactor expansion:
Determinant = (5(4) - (-3)(-1))(4 - (-1)) - (2(4) - (-3)(1))(2 - (-1)) + (2(2) - 5(1))(2 - 2)
= (20-3)(5) - (8+3)(3) + (4-5)(0)
= 17(5) - 11(3) + (-1)(0)
= 85 - 33 + 0
= 52
Since the determinant is non-zero (52), the system has one unique solution. Therefore, the correct answer for the first system of equations is A.) One Solution.
For the second system of equations:
2.) 2x + y - 3z = 4
4x + 2z = 10
-2x + 3y - 13z = -8
Similarly, find the determinant of the corresponding matrix:
| 2 1 -3 |
| 4 0 2 |
| -2 3 -13 |
Using cofactor expansion:
Determinant = (2(0) - 4(2))(3 - (-2)) - (4(0) - (-2)(3))(-2 - (-2)) + (4(3) - 2(-2))(2 - (-2))
= (-8)(5) - (-6)(0) + (12+4)(4)
= -40 - 0 + 64
= 24
Since the determinant is non-zero (24), the system has one unique solution. Therefore, the correct answer for the second system of equations is A.) One Solution.
So the answers you provided are correct:
1.) C.) No Solution
2.) A.) One Solution