Unit 2: Solving Systems of Linear Equations
Two ways to picture a system of equations, and Gaussian elimination as the mechanical process that solves either picture.
12 topics · 44 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
The Row Picture of a Linear System
- How it worksHow does each equation in a system draw one line, with the solution sitting at their intersection?
- Worked exampleFind the solution of the system x + y = 3 and x - y = 1 by graphing both lines.
- Common mistakeWhy doesn't a system of two equations always have exactly one crossing point?
- Going furtherWhat happens to the row picture when the two lines drawn by the equations are parallel?
Your turn1 of 2
Solve the system x + y = 4 and x - y = 3 for x.The Column Picture of a Linear System
- How it worksHow does writing Ax = b turn a system into one vector equation about combining columns?
- Worked exampleFind how the system 2x + y = 5, x - y = 1 can be written as a combination of columns equal to a vector b.
- Common mistakeWhy doesn't the column picture change the actual solution to a system, even though it looks completely different?
- Ties togetherHow does the column picture connect a system of equations back to the span of a set of vectors?
Your turn1 of 2
Write the system 2x + y = 4, x - y = 3 as a combination of the columns (2, 1) and (1, -1) equaling (4, 3). Find x.Gaussian Elimination, Step by Step
- How it worksHow does eliminating x from every equation below the first row simplify the system one variable at a time?
- Worked exampleSolve the system x + 2y = 5, 3x + y = 5 using elimination.
- Worked exampleSolve the system 2x + y - z = 3, x - y + 2z = 4, 3x + y + z = 8 using elimination.
- Common mistakeWhy doesn't the order you eliminate variables in ever change the final solution?
Your turn1 of 2
Solve the system x + 2y = 4, 3x + y = 3 using elimination. Find x.Elimination Using Matrices
- How it worksHow does multiplying by an elimination matrix subtract a multiple of one row from another?
- Worked exampleFind the elimination matrix that subtracts 2 times row 1 from row 2 of a 2 by 2 matrix.
- Going furtherWhat happens when the same elimination matrix is applied a second time in a row?
- Common mistakeWhy doesn't multiplying by an elimination matrix ever change what a system's solution set actually is?
Your turn1 of 2
The elimination matrix E subtracts 5 times row 1 from row 2. Applying E to the matrix [[2, 7], [-4, -7]] changes its (2,1) entry. Find the new (2,1) entry.Row Echelon Form
- How it worksHow does each pivot sit strictly to the right of the pivot in the row directly above it?
- Worked exampleFind the row echelon form of the matrix [[1, 2, 1], [2, 4, 3], [1, 1, 1]].
- Common mistakeWhy isn't a matrix in echelon form just any matrix that happens to have a lot of zeros?
- Going furtherWhat happens to a matrix's echelon form when two of its rows start out identical?
Your turn1 of 2
Perform one elimination step on the matrix [[1, 3, 2], [7, -4, -7], [-5, 3, -8]]: subtract 7 times row 1 from row 2. Find the new (2,2) entry.Reduced Row Echelon Form
- How it worksHow does continuing row operations upward clear out every entry above each pivot?
- Worked exampleFind the reduced row echelon form of the matrix [[1, 2, 3], [2, 5, 8]].
- Ties togetherHow does the reduced row echelon form of a matrix reveal its rank at a single glance?
Your turn1 of 2
Find the (1,3) entry of the reduced row echelon form obtained by subtracting 3 times row 2 from row 1 of [[1, 3, 2], [0, 1, 7]].Pivots and Free Variables
- How it worksHow does a column with no pivot in it mark a free variable in the solution?
- Worked exampleFind the pivot variables and free variables in the system reduced to [[1, 0, 2], [0, 1, -1]].
- Common mistakeWhy doesn't having more variables than equations automatically mean there is no solution?
- Going furtherWhat happens to the number of free variables the moment a row of all zeros shows up during elimination?
Your turn1 of 2
A system in 7 unknowns reduces to row echelon form with 2 pivot variables. Find the number of free variables.Systems with No Solution or Infinitely Many
- How it worksHow does a row that reduces down to 0 = 5 during elimination prove a system has no solution?
- Worked exampleShow that the system x + y = 2, 2x + 2y = 7 has no solution.
- Worked exampleShow that the system x + y = 2, 2x + 2y = 4 has infinitely many solutions.
- Common mistakeWhy doesn't a system with infinitely many solutions mean the original equations were wrong?
Your turn1 of 2
The system x + y = 4, 2x + 2y = q has either no solution or infinitely many, depending on q. Find the value of q that makes the system consistent (infinitely many solutions).Systems with More Equations than Unknowns
- How it worksHow does elimination on an overdetermined system typically produce a row that reveals no exact solution exists?
- Worked exampleShow that the overdetermined system x = 1, x = 2, 2x = 2 has no exact solution.
- Ties togetherHow does an overdetermined system with no exact solution set up the least squares problem solved later on?
Your turn1 of 2
The overdetermined system x = 4, x = 3, x = 9 has no exact solution unless a = b = c. Find the least-squares best estimate of x — the value minimizing the sum of squared errors.Systems with More Unknowns than Equations
- How it worksHow do the leftover unknowns become free variables that can be set to absolutely anything?
- Worked exampleSolve the underdetermined system x + y + z = 6 for all of its solutions.
- Common mistakeWhy can't an underdetermined system ever settle on just one single unique solution?
Your turn1 of 2
The underdetermined system x + y + z = 11 is solved by setting y and z freely. If y = 3 and z = 9, find x.Back Substitution
- How it worksHow does knowing the last variable let you solve for every variable above it, one at a time?
- Worked exampleSolve the triangular system x + y + z = 6, y + 2z = 5, 3z = 3 using back substitution.
- Going furtherWhat happens to back substitution when one of the pivots in the triangular system turns out to be zero?
Your turn1 of 2
A triangular system reduces to: 5 times z = 10; y + 2 times z = -4; x + y + z = -8. Solve by back substitution. Find x.Singular Systems and Elimination Breaking Down
- How it worksHow does a zero showing up in the pivot position force a row swap or expose a singular matrix?
- Worked exampleShow what happens when you try to eliminate on the matrix [[0, 1], [1, 1]] without swapping rows first.
- Common mistakeWhy doesn't hitting a zero pivot always mean the system has no solution?
- Going furtherWhat happens to elimination once an entire row of a matrix turns to zero partway through?
Your turn1 of 2
The matrix [[0, 3, 2], [6, -4, -7], [-5, 3, -8]] has a zero in the (1,1) pivot position. After swapping rows 1 and 2, eliminate row 3 using the new row 1. Find the resulting (3,2) entry.