1.solve the set of linear equation by matrix method.a+3b+2c=3,2a-b-3c=-8,5a+2b+c=9 solve for a,b,c
2.solve by Guassian elimination method,
(a)a+2b+3c=5,3a-b+2c=8,4a-6b-4c=-2,
(b)
a=2
b=−3
c=5
that's ur answer for a.b.c
Yes
a=2,b=5,c=3
To solve a system of linear equations by the matrix method, we can write the given equations in matrix form and then use matrix operations to find the values of the variables.
1. Given equations:
a + 3b + 2c = 3
2a - b - 3c = -8
5a + 2b + c = 9
Step 1: Write the augmented matrix:
| 1 3 2 | 3 |
| 2 -1 -3 | -8 |
| 5 2 1 | 9 |
Step 2: Perform row operations to simplify the matrix.
R2 = R2 - 2R1
R3 = R3 - 5R1
| 1 3 2 | 3 |
| 0 -7 -7 | -14 |
| 0 -13 -9 | -6 |
Step 3: Perform further row operations.
R2 = R2 / -7
R3 = R2 + R3
| 1 3 2 | 3 |
| 0 1 1 | 2 |
| 0 0 -2 | -4 |
Step 4: Solve for the variables.
From the last row, we have -2c = -4, which gives us c = 2.
Substituting c = 2 into the second row, we have b + 1 = 2, which gives us b = 1.
Substituting b = 1 and c = 2 into the first row, we have a + 3 + 4 = 3, which gives us a = -4.
Therefore, the solutions are: a = -4, b = 1, c = 2.
2. Gaussian elimination method involves manipulating a system of equations using row operations to simplify it and find the values of the variables. Let's solve the given equations using Gaussian elimination method.
(a) Given equations:
a + 2b + 3c = 5
3a - b + 2c = 8
4a - 6b - 4c = -2
Step 1: Write the augmented matrix:
| 1 2 3 | 5 |
| 3 -1 2 | 8 |
| 4 -6 -4 | -2 |
Step 2: Perform row operations to simplify the matrix.
R2 = R2 - 3R1
R3 = R3 - 4R1
| 1 2 3 | 5 |
| 0 -7 -7 | -7 |
| 0 -14 -16 | -22 |
Step 3: Perform further row operations.
R2 = R2 / -7
R3 = R2 * 2 + R3
| 1 2 3 | 5 |
| 0 1 1 | 1 |
| 0 0 -2 | -20 |
Step 4: Solve for the variables.
From the last row, we have -2c = -20, which gives us c = 10.
Substituting c = 10 into the second row, we have b + 10 = 1, which gives us b = -9.
Substituting b = -9 and c = 10 into the first row, we have a + 2(-9) + 3(10) = 5, which gives us a = -2.
Therefore, the solutions are: a = -2, b = -9, c = 10.
(b) Please provide the equations for part (b) to solve using Gaussian elimination method.