 5.2.1: The system of equations describes the distance of two hikers, Edna ...
 5.2.2: Check that each ordered pair is a solution to each system. If the p...
 5.2.3: Solve each equation by symbolic manipulation. a.14 + 2x = 4 3x b. 7...
 5.2.4: Solve the system of equations using the substitution method, and ch...
 5.2.5: Substitute 4 3x for y. Then rewrite each expression in terms of one...
 5.2.6: Solve each system of equations by substitution, and check your solu...
 5.2.7: APPLICATION This system of equations models the profits of two home...
 5.2.8: The costs for two families to attend Friday nights basketball game ...
 5.2.9: APPLICATION The manager of a movie theater wants to know the number...
 5.2.10: Students in an algebra class did an experiment similar to the Inves...
 5.2.11: The table at right gives the equations that model the three vehicle...
 5.2.12: APPLICATION This table shows the winning times for the Olympic wome...
 5.2.13: A candy store manager is making a sour candy mix by combining sour ...
 5.2.14: Mrs. Abdul mixes bottled fruit juice with natural orange soda to ma...
 5.2.15: A system of two linear equations has the solution (3, 4.5).Write th...
 5.2.16: You and your family are visiting Seattle and take the elevator to t...
 5.2.17: Do each calculation by hand, and then check your results with a cal...
 5.2.18: Match each matrix multiplication with its answer.
Solutions for Chapter 5.2: Solving Systems of Equations Using Substitution
Full solutions for Discovering Algebra: An Investigative Approach  2nd Edition
ISBN: 9781559537636
Solutions for Chapter 5.2: Solving Systems of Equations Using Substitution
Get Full SolutionsDiscovering Algebra: An Investigative Approach was written by Patricia and is associated to the ISBN: 9781559537636. Chapter 5.2: Solving Systems of Equations Using Substitution includes 18 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2. Since 18 problems in chapter 5.2: Solving Systems of Equations Using Substitution have been answered, more than 3075 students have viewed full stepbystep solutions from this chapter.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Solvable system Ax = b.
The right side b is in the column space of A.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
I don't want to reset my password
Need help? Contact support
Having trouble accessing your account? Let us help you, contact support at +1(510) 9441054 or support@studysoup.com
Forgot password? Reset it here