 3.1.1: Solve each system of equations by completing a table.
 3.1.2: Solve each system of equations by completing a table.
 3.1.3: Solve each system of equations by graphing.
 3.1.4: Solve each system of equations by graphing.
 3.1.5: Write equations that represent the cost of printing digital photos ...
 3.1.6: Under what conditions is the cost to print digital photos the same ...
 3.1.7: When is it best to use EZ Online Digital Photos and when is it best...
 3.1.8: When is it best to use EZ Online Digital Photos and when is it best...
 3.1.9: When is it best to use EZ Online Digital Photos and when is it best...
 3.1.10: When is it best to use EZ Online Digital Photos and when is it best...
 3.1.11: Solve each system of linear equations by completing a table.
 3.1.12: Solve each system of linear equations by completing a table.
 3.1.13: Solve each system of linear equations by graphing.
 3.1.14: Solve each system of linear equations by graphing.
 3.1.15: Solve each system of linear equations by graphing.
 3.1.16: Solve each system of linear equations by graphing.
 3.1.17: Solve each system of linear equations by graphing.
 3.1.18: Solve each system of linear equations by graphing.
 3.1.19: Graph each system of equations and describe it as consistent and in...
 3.1.20: Graph each system of equations and describe it as consistent and in...
 3.1.21: Graph each system of equations and describe it as consistent and in...
 3.1.22: Graph each system of equations and describe it as consistent and in...
 3.1.23: Graph each system of equations and describe it as consistent and in...
 3.1.24: Graph each system of equations and describe it as consistent and in...
 3.1.25: The sides of an angle are parts of two lines whose equations are 2y...
 3.1.26: The graphs of y  2x = 1, 4x + y = 7, and 2y  x = 4 contain the s...
 3.1.27: If the price for vitamins is $8.00 a bottle, what is the supply of ...
 3.1.28: If the price for vitamins is $12.00 a bottle, what is the supply of...
 3.1.29: At what quantity will the prices stabilize? What is the equilibrium...
 3.1.30: Write equations that represent the populations of Florida and New Y...
 3.1.31: Graph both equations for the years 2003 to 2020. Estimate when the ...
 3.1.32: Do you think New York will overtake Texas as the second most populo...
 3.1.33: Solve each system of equations by graphing.
 3.1.34: Solve each system of equations by graphing.
 3.1.35: Solve each system of equations by graphing.
 3.1.36: Graph each system of equations and describe it as consistent and in...
 3.1.37: Graph each system of equations and describe it as consistent and in...
 3.1.38: Graph each system of equations and describe it as consistent and in...
 3.1.39: To use a TI83/84 Plus to solve a system of equations, graph the eq...
 3.1.40: To use a TI83/84 Plus to solve a system of equations, graph the eq...
 3.1.41: To use a TI83/84 Plus to solve a system of equations, graph the eq...
 3.1.42: Give an example of a system of equations that is consistent and ind...
 3.1.43: Explain why a system of linear equations cannot have exactly two so...
 3.1.44: State the conditions for which the system below is: (a) consistent ...
 3.1.45: Use the information about sales on page 116 to explain how a system...
 3.1.46: Which of the following best describes the graph of the equations? 4...
 3.1.47: Which set of dimensions corresponds to a triangle similar to the on...
 3.1.48: Simon is putting up fence around his yard at a rate no faster than ...
 3.1.49: Identify each function as S for step, C for constant, A for absolut...
 3.1.50: Identify each function as S for step, C for constant, A for absolut...
 3.1.51: Identify each function as S for step, C for constant, A for absolut...
 3.1.52: Simplify each expression.
 3.1.53: Simplify each expression.
 3.1.54: Simplify each expression.
 3.1.55: Simplify each expression.
 3.1.56: Simplify each expression.
 3.1.57: Simplify each expression.
Solutions for Chapter 3.1: Solving Systems of Equations by Graphing
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving  1st Edition
ISBN: 9780078778568
Solutions for Chapter 3.1: Solving Systems of Equations by Graphing
Get Full SolutionsChapter 3.1: Solving Systems of Equations by Graphing includes 57 full stepbystep solutions. This textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1. California Algebra 2: Concepts, Skills, and Problem Solving was written by and is associated to the ISBN: 9780078778568. This expansive textbook survival guide covers the following chapters and their solutions. Since 57 problems in chapter 3.1: Solving Systems of Equations by Graphing have been answered, more than 44113 students have viewed full stepbystep solutions from this chapter.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

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.

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)·(b  Ax) = o.

Outer product uv T
= column times row = rank one matrix.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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).

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.