Practical - problem set 2 PDF

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1 Practice problems List 2

MATH1119B

Practicing is very important in order to do well in this course. Practice as many problems as possible until you feel confortable with the material. Solutions to odd-value problems of the textbook can be found at the end of the textbook. Solutions to some questions that are not in the textbook will be posted on cuLearn later. 1. Determine which of the following equations are linear. a) 8x1 − 7x2 + x3 = −1 b) −2x + 8y + z = −5 c) d) 9x1 −

2 x2

√

5x1 − 34x2 + 73 x3 = 1

√ √ + 9x3 − x4 = −6 e) 9x − 5 y + 12z = 2 e) 8x1 x3 + 7x2 x3 − 9x3 = 10

f) 4x2 − 8xy + 5z = 1 2. Determine if the given point is a solution to the given equation. a) Is (−3, 1, 0) a solution of 4x1 + 7x2 + 9x3 = −5 ? b) Is (−1, 2, 4) a solution of 4x1 + 7x2 + 9x3 = −5 ? c) Is (−1, 2, 3, 4, −2) a solution of 2x1 − x2 + 3x3 + x4 + 2x5 = 4 ? d) Is (3, 2, 0, −2, 3) a solution of 2x1 − x2 + 3x3 + x4 + 2x5 = 4 ? 3. Determine if the given point is a solution to the given linear system of equation. a) Is (2, −1, 3) a solution of the linear system 2x1 + 5x2 + 3x3 = 8 x1 − 2x2 + x3 = 7 4x1 + 6x2 + x3 = 5 b) Is (2, 1, −2, 4) a solution of the linear system x1 + x2 + 2x3 + x4 = 3 2x1 − x2 − x3 − x4 = 1

x3 + x4 = 2 4x2 + x3 − x4 = −2

2 c) Is (1, 3, 4) a solution of the linear system 4x1 + 2x2 − 3x3 = −2 2x1 − x2 + 4x3 = 15 x1 − x2 + 2x3 = 7

4. a) Find the augmented matrix of the linear system 3x1 + 2x2 − 4x4 = 0 −2x2 + 3x3 − x4 = −2 −5x2 + 7x3 = 1

b) Find a linear system of equations for which the given matrix is the augmented matrix of the system. Use the variables x1 , x2 , x3 .   8 2 6 1 2 9 4 −6   1 −3 −6 −2 5 −7 4 4

5. Use Gaussian elimination (and later you can try to use Gauss-Jordan elimination) to solve the following linear systems that all have a unique solution. x1 + 2x2 = 8 a) 3x1 − 4x2 = 4

2x1 − 3x2 + 4x3 = −12 = −5 b) x1 − 2x2 + x3 3x1 + x2 + 2x3 =1

x1 + x2 = 1 d) 2x1 − x2 = 5 3x1 + 4x2 = 2

2x1 + 3x2 = 13 e) x1 − 2x2 = 3 . 5x1 + 2x2 = 27

3x1 + 2x2 + x3 = 2 c) 4x1 + 2x2 + 2x3 = 8 x1 − x2 + x3 =4

3 6. Use Gaussian elimination (and later you can try to use Gauss-Jordan elimination) to solve the following linear systems that all have infinitely many solutions (make sure to see why). Write the parametric equation that gives all the solutions. x1 − x2 − x3 + 2x4 =1 x1 + 2x2 − x3 b) a) 2x1 − 2x2 − x3 + 3x4 = 3 2x1 + 3x2 + x3 −x1 + x2 − x3 = −3 x1 − 3x2 − 2x3 = 0 2x1 + 3x2 − x3 + 4x4 c) −x1 + 2x2 + x3 = 0 d) 3x1 − x2 + x4 2x1 + 4x2 + 6x3 = 0 3x1 − 4x2 + x3 − x4

=3 =1 =0 =1 =2

7. Use Gaussian elimination (and later you can try to use Gauss-Jordan elimination) to solve the following linear systems that all have no solution (make sure x1 − x2 + 2x3 = 3 x1 − x2 + x3 =0 to understand why). a) x1 + 2x2 − x3 = −3 b) −x1 + 3x2 + x3 = 5 c) 2x2 − 2x3 =1 3x1 + x2 + 7x3 = 2 x1 + x2 + 2x3 + x4 = 1 x1 + x2 =5 x1 − x2 − x3 + x4 = 0 d) x2 + x3 = −1 3x1 + 3x2 = 10 x1 + x2 + x4 =2 8. If you want more practice with solving linear systems. Try solving the following linear systems using either Gaussian elimination or Gauss-Jordan elimination. Indicate all the steps that you used to get to your final answer. (Note: This problem contains linear systems with no solution, a unique solution and infinitely many solutions. It is up to you to realize which one it is.) a)

x1 − 2x2 − 3x3 = 0 4x2 − 8x3 = 28 2x1 + 2x2 − 3x3 + x4 = 11 x2 + x3 = 1 b) 2x1 + 8x2 − 2x3 = 12 c) x1 + x2 + x3 + x4 =4 x1 + x2 − x3 = 1 −3x1 − 10x2 = −6 2x1 − 2x2 + 2x3 + 3x4 = 7

2x − 2z y+z d) 2x + y − z 3y + 3z

=6 =1 =7 =0

x1 + 3x2 + 5x3 + 3x4 = −24 2x1 + 7x2 + 11x3 + 7x4 = −54 e) −x1 − x2 − 7x4 = −6 −2x1 − 9x2 − 14x3 − 5x4 = 74

2x2 + 3x3 =8 f) 2x1 + 3x2 + x3 = 5 x1 − x2 − 2x3 = −5

x1 + 2x2 − 3x3 = 9 g) 2x1 − x2 + x3 = 0 4x1 − x2 + x3 = 4

2x1 + x2 =3 h) 4x1 + x2 =7 2x1 + 5x2 = −1

4

i)

−x1 + 3x2 − 2x3 + 4x4 = 0 2x1 − 6x2 + x3 − 2x4 = −3 x1 − 3x2 + 4x3 − 8x4 =2

l)

3x1 + 4x2 − x3 = 8 6x1 + 8x2 − 2x3 = 3

x1 + 3x2 = −4 o) 2x1 + 5x2 = −8 x1 + 3x2 = −5

m)

j)

x1 + x2 − 2x3 = 5 2x1 + 3x2 + 4x3 = 2

x1 + x2 + 3x3 = 12 2x1 + 2x2 + 6x3 = 6

2x1 + 3x2 − x3 = 6 p) 2x1 − x2 + 2x3 = −8 3x1 − x2 + x3 = −7

k)

x1 + 4x2 − x3 = 12 3x1 + 8x2 + 4x3 = 4

x1 − 5x2 = 6 n) 3x1 + 2x2 = 1 5x1 + 2x2 = 1

2x1 + 4x2 + 6x3 = −12 q) 2x1 − 3x2 − 4x3 = 15 3x1 + 4x2 + 5x3 = −8

9. Use the Gauss-Jordan elimination method (and only that method) to solve the following linear systems: a)

x1 − x2 + x3 = 0 x2 − 2x3 =1 2x1 − x2 =1

b)

2x1 − 3x2 + x3 = 1 −x1 + 2x3 =0 3x1 − 3x2 − x3 = 1

c)

x2 − 2x3 + x4 =1 2x1 − x2 − x4 =0 4x1 + x2 − 6x3 + x4 = 3...