Please help and I would appreciate an explanation as to how you got to the answer.
1.) Solve the system by triangularizing the augmented matrix and using back substitution.
0.09x-0.15y+0.39
-0.18x+0.35y=-0.88
2.) Solve the system by triangularizing the augmented matrix and using back substitution.
-x-y+z=1
x-y-4z=-7
4x+y+z=6
3.) Perform the indicated row operation, then write the new matrix.
-4 4 | 4
9 -6 | 5 -2R1+ R2 ---> R2
Sure! Let's go step by step for each problem.
1.) Solve the system by triangularizing the augmented matrix and using back substitution.
The given system is:
0.09x - 0.15y = 0.39 ...(1)
-0.18x + 0.35y = -0.88 ...(2)
To triangularize the augmented matrix, we need to eliminate the variable x in the second equation (2). We can do this by multiplying equation (1) by 2 and adding it to equation (2). The resulting equation will have the x variable eliminated:
Multiply equation (1) by 2:
0.18x - 0.30y = 0.78
Add equation (1) (multiplied by 2) to equation (2):
0.18x - 0.30y + (-0.18x + 0.35y) = 0.78 + (-0.88)
Simplifying the equation:
0.05y = -0.10
Divide both sides of the equation by 0.05:
y = -2
Now that we have the value of y, we can substitute it back into equation (1) and solve for x:
0.09x - 0.15(-2) = 0.39
0.09x + 0.30 = 0.39
0.09x = 0.39 - 0.30
0.09x = 0.09
Divide both sides of the equation by 0.09:
x = 1
So, the solution to the system is x = 1, y = -2.
2.) Solve the system by triangularizing the augmented matrix and using back substitution.
The given system is:
-x - y + z = 1 ...(1)
x - y - 4z = -7 ...(2)
4x + y + z = 6 ...(3)
To triangularize the augmented matrix, we'll use the method of elimination:
Starting with equation (1), add it to equation (2) to eliminate the x variable:
(-x - y + z) + (x - y - 4z) = 1 + (-7)
-z = -6
Multiply the equation by -1:
z = 6
Substitute the value of z back into equation (1):
-x - y + 6 = 1
Simplify the equation:
-x - y = -5
Now we have a system of two equations:
-x - y = -5 ...(4)
4x + y = 5 ...(5)
To eliminate the y variable, multiply equation (4) by 4 and add it to equation (5):
(-4x - 4y) + (4x + y) = (-5) + 5
-3y = 0
Divide both sides of the equation by -3:
y = 0
Now that we have the value of y, we can substitute it back into equation (4) and solve for x:
-x - 0 = -5
-x = -5
Multiply both sides of the equation by -1:
x = 5
So, the solution to the system is x = 5, y = 0, z = 6.
3.) Perform the indicated row operation, then write the new matrix.
The given matrix is:
-4 4 | 4
9 -6 | 5
To perform the row operation -2R1 + R2, we multiply the first row by -2 and add it to the second row:
Multiply the first row (-4 4 | 4) by -2:
(-2)(-4) (-2)(4) (-2)(4) = 8 -8 -8
Add the resulting row to the second row:
8 -8 -8 + 9 -6 5 = 17 -14 -3
The new matrix after performing the row operation is:
-4 4 | 4
17 -14 -3
I hope this explanation helps! Let me know if you have any further questions.