Question

Question 1 (5 points): Consider the system of linear equations x1 − 4x2 − x4 = −7 x2 − 2x4 = 3 x4 + 2x5 = 3 (a) Write down the coefficient matrix of this system. (b) Write down the augmented matrix of this system. (c) Use row operations to put the augmented matrix into reduced echelon form. Make it clear which row operations you are using. (d) Is the system consistent or inconsistent? Explain your answer. (e) If the system is consistent, use your answer to part (c) to find the solution to the linear system.

Answer 1

Answer:

Step-by-step explanation:

Given is a system of equations as

$x1-4x2 - x4 = -7\\ \\x2 - 2x4 = 3 \\\\x4 + 2x5 = 3$

We have 5 variables and 3 equations

a) coefficient matrix of this system is

1   -4   0   -1   0\\

0    1   0   -2   0\\

0    0   0    1    2\\

We find that x3 has no coefficient in any of the equations so we can omit x3 and write as equations for 4 variables as

1   -4     -1   0\\

0    1     -2   0\\

0    0     1    2\\

b) Augmented matrix is

1   -4     -1   0\\  7

0    1     -2   0\\ 3

0    0     1    2\\3

c) For row operations to ehelon form

we can do R1+4R2 = R1

We get

1   0   -9  0 \\  19

0   1  -2   0 \\   3

0   0   1   2  \\   3

Now let us do R1 = R1+9R3 and R2 = R2+2R3

1  0   0   0  \\ 46

0  1   0   0   \\  9

0   0  1   2   \\  3

d) We find that there are infinite solutions to  the system in parametric form, since x4 and x5 are linked with only one equation

e) x1 = 46, x2 = 9, x4+2x5 =3

Or x1 =46, x2 =9, x4 = 3-2x5, x5 = x5 is the parametric solution