Deck 1: Systems of Linear Equations

Question 1

Determine which of the points (2, -3), (2, 3), and (4, 2) lie on both of the lines
and

.

Answer 1

(2, 3)

Question 2

Determine which of the points (1, 0, -1, 0), (0, 1, 2, 3), and (2, 1, -1, -1) satisfy the linear system

Answer 2

(0, 1, 2, 3)

Question 3

Determine which of (3, s₂, s₁, 2), (0, 0, 0 ,0), (2 - s₁ - s₂, 1 + s₁ + s₂, s₁, s₂), and (3 - s₁, s₁, s₂, 2 - s₂) form a solution to the following system for all choices of the free parameters

.

Answer 3

(2 - s₁ - s₂, 1 + s₁ + s₂, s₁, s₂)

Question 4

Determine if the system is in echelon form, and if so, identify the leading variables and the free variables. If it is not in echelon form, explain why.

Question 5

Determine if the system is in echelon form, and if so, identify the leading variables and the free variables. If it is not in echelon form, explain why.

Question 6

Find all solutions to the system

Question 7

Find all solutions to the system

Question 8

Find all solutions to the system

Question 9

Reorder the equations to put the following system of three equations with four unknowns in echelon form:
​

Question 10

Determine the value(s) of
so that the following system is consistent.​

Question 11

Suppose a system of
equations has two free variables. How many leading variables are there?

Question 12

If a linear system has more variables than equations, then the system is inconsistent.

Question 13

If a linear system has infinitely many solutions, then there are more variables than equations.

Question 14

If a linear system has no free variables, then there exists at most one solution.

Question 15

Convert the augmented matrix to the equivalent linear system.

Question 16

Convert the augmented matrix to the equivalent linear system.

Question 17

Determine whether the matrix is in reduced echelon form, echelon form only, or not in echelon form.

Question 18

Determine whether the matrix is in reduced echelon form, echelon form only, or not in echelon form.

Question 19

Determine whether the matrix is in reduced echelon form, echelon form only, or not in echelon form.

Question 20

Identify the row operation which transforms the matrix on the left to the matrix on the right.

Question 21

Identify the row operation which transforms the matrix on the left to the matrix on the right.

Question 22

Convert the given system to an augmented matrix, and find all solutions by reducing to echelon form and using back substitution.

Question 23

Convert the given system to an augmented matrix, and find all solutions by reducing to echelon form and using back substitution.

Question 24

Convert the given system to an augmented matrix, and find all solutions by reducing to echelon form and using back substitution , if needed.

Question 25

Convert the given system to an augmented matrix, and find all solutions by transforming to reduced echelon form and using back substitution.

Question 26

Convert the given system to an augmented matrix, and find all solutions by transforming to reduced echelon form and using back substitution.

Question 27

Every homogeneous linear system has at least one solution.

Question 28

A linear system with infinitely many solutions must have more variables than equations.

Question 29

A linear system with a unique solution can not have more variables than equations.

Question 30

The volume of traffic for a collection of intersections is shown. Find all possible values for

,

,

, and

.

Question 31

The volume of traffic for a collection of intersections is shown. Find the minimum volume of traffic from C to D.

Question 32

Find all possible equilibrium and endpoint temperatures for the heavy wires with endpoints held at the indicated temperatures.

Question 33

Find the equilibrium temperatures for the heavy wires with endpoints held at the given temperatures.

Question 34

An economy has three industries: A, B, and C. These industries have annual consumer sales of 45, 37, and 64 (in millions of dollars), respectively. In addition, for every dollar of goods that A sells, A requires 25 cents from B and 15 cents from C. For each dollar of goods that B sells, B requires 35 cents from A and 25 cents from C. For each dollar of goods that C sells, C requires 20 cents from A and 45 cents from B. Let a, b, c be the total output from industries A, B, C, respectively. What values of a, b, c (rounded to the nearest thousand dollars) will satisfy both consumer and between-industry demand?

Question 35

Balance the given chemical equation.

Question 36

Balance the given chemical equation.

Question 37

Use a system of linear equations to find the values
and
for the partial fraction decomposition

Question 38

Use a system of linear equations to find the values

,

, and
for the partial fraction decomposition

Question 39

The points (-6, 0, -1), (3, 2, 0), and (0, 3, -1) lie on a plane ax + by + cz = 1. Use a system of linear equations to find the equation of this plane.

Question 40

Use a system of linear equations to find the equation
of the parabola which passes through the points (1, 2), (2, 0), and (3, -4).

Question 41

Use a system of linear equations to find a function of the form
such that

,

,

, and

.

Question 42

Use a system of linear equations to find the values of the coefficients a, b, c if
with

,

, and

.

Question 43

Use partial pivoting with Gaussian elimination to find the solutions to the system.

Question 44

Use partial pivoting with Gaussian elimination to find the solutions to the system.

Question 45

Solve the given system using Gaussian elimination with three significant digits of accuracy. Then solve the system again, incorporating partial pivoting.

Question 46

Solve the given system using Gaussian elimination with three significant digits of accuracy. Then solve the system again, incorporating partial pivoting.

Question 47

Compute the first three Jacobi iterations, using
as the initial value for each variable. Then find the exact solution and compare.

Question 48

Compute the first three Jacobi iterations, using
as the initial value for each variable. Then find the exact solution and compare.

Question 49

Compute the first three Gauss-Seidel iterations for the system in question 5, using
as the initial value for each variable. Then find the exact solution and compare.

Question 50

Compute the first three Gauss-Seidel iterations for the system in question 6, using
as the initial value for each variable. Then find the exact solution and compare.

Question 51

Determine if the system is diagonally dominant. If not, then if possible rewrite the system so that it is diagonally dominant.

Question 52

Compute the first four Jacobi iterations for the system as written, with the initial value of each variable set equal to

. Then rewrite the system so that it is diagonally dominant, set the value of each variable to 0, and again compute four Jacobi iterations. Finally, find the exact solution and compare.

Question 53

Compute the first four Jacobi iterations for the system as written, with the initial value of each variable set equal to

. Then rewrite the system so that it is diagonally dominant, set the value of each variable to 0, and again compute four Jacobi iterations. Finally, find the exact solution and compare.

Question 54

Compute the first four Gauss-Seidel iterations for the system in question 10, with the initial value of each variable set equal to 0. Then rewrite the system so that it is diagonally dominant, set the value of each variable to 0, and again compute four Gauss-Seidel iterations.

Question 55

Compute the first four Gauss-Seidel iterations for the system in question 11, with the initial value of each variable set equal to

. Then rewrite the system so that it is diagonally dominant, set the value of each variable to 0, and again compute four Gauss-Seidel iterations.

Question 56

The values for the first few Jacobi iterations for a linear system are given. Find the values for the next iteration.

Question 57

The values for the first few Gauss-Seidel iterations for a linear system are given. Find the values for the next iteration.