 51.1: Use the graph to determine whether each system has no solution, one...
 51.2: Use the graph to determine whether each system has no solution, one...
 51.3: Use the graph to determine whether each system has no solution, one...
 51.4: Use the graph to determine whether each system has no solution, one...
 51.5: Graph each system of equations. Then determine whether the system h...
 51.6: Graph each system of equations. Then determine whether the system h...
 51.7: Graph each system of equations. Then determine whether the system h...
 51.8: Graph each system of equations. Then determine whether the system h...
 51.9: GEOMETRY The length of the rectangle is 1 meter less than twice its...
 51.10: Use the graph to determine whether each system has no solution, one...
 51.11: Use the graph to determine whether each system has no solution, one...
 51.12: Use the graph to determine whether each system has no solution, one...
 51.13: Use the graph to determine whether each system has no solution, one...
 51.14: Use the graph to determine whether each system has no solution, one...
 51.15: Use the graph to determine whether each system has no solution, one...
 51.16: Graph each system of equations. Then determine whether the system h...
 51.17: Graph each system of equations. Then determine whether the system h...
 51.18: Graph each system of equations. Then determine whether the system h...
 51.19: Graph each system of equations. Then determine whether the system h...
 51.20: Graph each system of equations. Then determine whether the system h...
 51.21: Graph each system of equations. Then determine whether the system h...
 51.22: Graph each system of equations. Then determine whether the system h...
 51.23: Graph each system of equations. Then determine whether the system h...
 51.24: Graph each system of equations. Then determine whether the system h...
 51.25: Graph each system of equations. Then determine whether the system h...
 51.26: Graph each system of equations. Then determine whether the system h...
 51.27: Graph each system of equations. Then determine whether the system h...
 51.28: SAVINGS For Exercises 28 and 29, use the following information. Mon...
 51.29: SAVINGS For Exercises 28 and 29, use the following information. Mon...
 51.30: BALLOONING For Exercises 30 and 31, use the information in the grap...
 51.31: BALLOONING For Exercises 30 and 31, use the information in the grap...
 51.32: Graph each system of equations. Then determine whether the system h...
 51.33: Graph each system of equations. Then determine whether the system h...
 51.34: ANALYZE GRAPHS For Exercises 3436, use the graph at the right.
 51.35: ANALYZE GRAPHS For Exercises 3436, use the graph at the right.
 51.36: ANALYZE GRAPHS For Exercises 3436, use the graph at the right.
 51.37: POPULATION For Exercises 3739, use the following information. The U...
 51.38: POPULATION For Exercises 3739, use the following information. The U...
 51.39: POPULATION For Exercises 3739, use the following information. The U...
 51.40: CHALLENGE The solution of the system of equations Ax + y = 5 and Ax...
 51.41: OPEN ENDED Write three equations such that they form a system of eq...
 51.42: REASONING Determine whether a system of two linear equations with (...
 51.43: Writing in Math Use the information on page 253 to explain how grap...
 51.44: A buffet restaurant has one price for adults and another price for ...
 51.45: REVIEW Francisco has 3 dollars more than _1 4 the number of dollars...
 51.46: Write the slopeintercept form of an equation for the line that pas...
 51.47: Write the slopeintercept form of an equation for the line that pas...
 51.48: PREREQUISITE SKILL Solve each equation for the variable specified.
 51.49: PREREQUISITE SKILL Solve each equation for the variable specified.
 51.50: PREREQUISITE SKILL Solve each equation for the variable specified.
Solutions for Chapter 51: Graphing Systems of Equations
Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1)  1st Edition
ISBN: 9780078738227
Solutions for Chapter 51: Graphing Systems of Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1) , edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Since 50 problems in chapter 51: Graphing Systems of Equations have been answered, more than 35428 students have viewed full stepbystep solutions from this chapter. Algebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227. Chapter 51: Graphing Systems of Equations includes 50 full stepbystep solutions.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Column space C (A) =
space of all combinations of the columns of A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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).

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.