 3.1.1: A solution to a system of linear equations in two variables is an o...
 3.1.2: When solving a system of linear equations by graphing, the systems ...
 3.1.3: When solving e 3x  2y = 5 y = 3x  3 by the substitution method, w...
 3.1.4: When solving e 2x + 10y = 9 8x + 5y = 7 by the addition method, we ...
 3.1.5: When solving e 4x  3y = 15 3x  2y = 10 by the addition method, we...
 3.1.6: When solving e 12x  21y = 24 4x  7y = 7 by the addition method, w...
 3.1.7: When solving e x = 3y + 2 5x  15y = 10 by the substitution method,...
 3.1.8: e x + y = 6 x  y = 4
 3.1.9: e 2x + y = 4 y = 4x + 1
 3.1.10: e x + 2y = 4 y = 2x  1
 3.1.11: e 3x  2y = 6 x  4y = 8
 3.1.12: e 4x + y = 4 3x  y = 3
 3.1.13: e 2x + 3y = 6 4x = 6y + 12
 3.1.14: e 3x  3y = 6 2x = 2y + 4
 3.1.15: e y = 2x  2 y = 5x + 5
 3.1.16: e y = x + 1 y = 3x + 5
 3.1.17: e 3x  y = 4 6x  2y = 4
 3.1.18: e 2x  y = 4 4x  2y = 6
 3.1.19: e 2x + y = 4 4x + 3y = 10
 3.1.20: e 4x  y = 9 x  3y = 16
 3.1.21: e x  y = 2 y = 1
 3.1.22: e x + 2y = 1 x = 3
 3.1.23: e 3x + y = 3 6x + 2y = 12
 3.1.24: e 2x  3y = 6 4x  6y = 24
 3.1.25: e x + y = 6 y = 2x
 3.1.26: e x + y = 10 y = 4x
 3.1.27: e 2x + 3y = 9 x = y + 2
 3.1.28: e 3x  4y = 18 y = 1  2x
 3.1.29: e y = 3x + 7 5x  2y = 8
 3.1.30: e x = 3y + 8 2x  y = 6
 3.1.31: e 4x + y = 5 2x  3y = 13
 3.1.32: e x  3y = 3 3x + 5y = 19
 3.1.33: e x  2y = 4 2x  4y = 5
 3.1.34: e x  3y = 6 2x  6y = 5
 3.1.35: e 2x + 5y = 4 3x  y = 11
 3.1.36: e 2x + 5y = 1 x + 6y = 8
 3.1.37: e 21x  12  y = 3 y = 2x + 3
 3.1.38: e x + y  1 = 21y  x2 y = 3x  1
 3.1.39: x 4  y 4 = 1 x + 4y = 9
 3.1.40: x 6  y 2 = 1 3 x + 2y = 3
 3.1.41: y = 2 5 x  2 2x  5y = 10
 3.1.42: y = 1 3 x + 4 3y = x + 12
 3.1.43: e x + y = 7 x  y = 3
 3.1.44: e 2x + y = 3 x  y = 3
 3.1.45: e 12x + 3y = 15 2x  3y = 13
 3.1.46: e 4x + 2y = 12 3x  2y = 16
 3.1.47: e x + 3y = 2 4x + 5y = 1
 3.1.48: e x + 2y = 1 2x  y = 3
 3.1.49: e 6x  y = 5 4x  2y = 6
 3.1.50: e x  2y = 5 5x  y = 2
 3.1.51: e 3x  5y = 11 2x  6y = 2
 3.1.52: e 4x  3y = 12 3x  4y = 2
 3.1.53: e 2x  5y = 13 5x + 3y = 17
 3.1.54: e 4x + 5y = 9 6x  3y = 3
 3.1.55: e 2x + 6y = 8 3x + 9y = 12
 3.1.56: e x  3y = 6 3x  9y = 9
 3.1.57: e 2x  3y = 4 4x + 5y = 3
 3.1.58: e 4x  3y = 8 2x  5y = 14
 3.1.59: e 3x  7y = 1 2x  3y = 1
 3.1.60: e 2x  3y = 2 5x + 4y = 51
 3.1.61: e x = y + 4 3x + 7y = 18
 3.1.62: e y = 3x + 5 5x  2y = 7
 3.1.63: 9x + 4y 3 = 5 4x  y 3 = 5
 3.1.64: x 6  y 5 = 4 x 4  y 6 = 2
 3.1.65: 1 4 x  1 9 y = 2 3 1 2 x  1 3 y = 1
 3.1.66: 1 16x  3 4 y = 1 3 4 x + 5 2 y = 11
 3.1.67: e x = 3y  1 2x  6y = 2
 3.1.68: e x = 4y  1 2x  8y = 2
 3.1.69: e y = 2x + 1 y = 2x  3
 3.1.70: e y = 2x + 4 y = 2x  1
 3.1.71: e 0.4x + 0.3y = 2.3 0.2x  0.5y = 0.5
 3.1.72: e 0.2x  y = 1.4 0.7x  0.2y = 1.6
 3.1.73: e 5x  40 = 6y 2y = 8  3x
 3.1.74: e 4x  24 = 3y 9y = 3x  1
 3.1.75: e 3(x + y) = 6 3(x  y) = 36
 3.1.76: e 4(x  y) = 12 4(x + y) = 2
 3.1.77: e 3(x  3)  2y = 0 2(x  y) = x  3
 3.1.78: e 5x + 2y = 5 4(x + y) = 6(2  x)
 3.1.79: e x + 2y  3 = 0 12 = 8y + 4x
 3.1.80: e 2x  y  5 = 0 10 = 4x  2y
 3.1.81: e 3x + 4y = 0 7x = 3y
 3.1.82: e 5x + 8y = 20 4y = 5x
 3.1.83: x + 2 2  y + 4 3 = 3 x + y 5 = x  y 2  5 2
 3.1.84: x  y 3 = x + y 2  1 2 x + 2 2  4 = y + 4 3
 3.1.85: e 5ax + 4y = 17 ax + 7y = 22
 3.1.86: e 4ax + by = 3 6ax + 5by = 8
 3.1.87: For the linear function f(x) = mx + b, f(2) = 11 and f(3) = 9. Fi...
 3.1.88: For the linear function f(x) = mx + b, f(3) = 23 and f(2) = 7. Fi...
 3.1.89: Write the linear system whose solution set is {(6, 2)}. Express eac...
 3.1.90: Write the linear system whose solution set is . Express each equati...
 3.1.91: We opened the section with a bar graph that showed fewer U.S. adult...
 3.1.92: The graph shows that from 2000 through 2006, Americans unplugged la...
 3.1.93: Although Social Security is a problem, some projections indicate th...
 3.1.94: The Rise and Fall of D and E The graph indicates that vitamin D sal...
 3.1.95: In this exercise, let x represent the number of years after 1985 an...
 3.1.96: In this exercise, let x represent the number of years after 1985 an...
 3.1.97: A chain of electronics stores sells handheld color televisions. Th...
 3.1.98: At a price of p dollars per ticket, the number of tickets to a rock...
 3.1.99: What is a system of linear equations? Provide an example with your ...
 3.1.100: What is a solution of a system of linear equations?
 3.1.101: Explain how to determine if an ordered pair is a solution of a syst...
 3.1.102: Explain how to solve a system of linear equations by graphing.
 3.1.103: Explain how to solve a system of equations using the substitution m...
 3.1.104: Explain how to solve a system of equations using the addition metho...
 3.1.105: When is it easier to use the addition method rather than the substi...
 3.1.106: When using the addition or substitution method, how can you tell if...
 3.1.107: When using the addition or substitution method, how can you tell if...
 3.1.108: Verify your solutions to any five exercises from Exercises 724 by u...
 3.1.109: Even if a linear system has a solution set involving fractions, suc...
 3.1.110: If I add the equations on the right and solve the resulting equatio...
 3.1.111: In the previous chapter, we developed models for life expectancy, y...
 3.1.112: Here are two models that describe winning times for the Olympic 400...
 3.1.113: The addition method cannot be used to eliminate either variable in ...
 3.1.114: The solution set of the system e 5x  y = 1 10x  2y = 2 is {(2, 9)}.
 3.1.115: A system of linear equations can have a solution set consisting of ...
 3.1.116: The solution set of the system e y = 4x  3 y = 4x + 5 is the empty...
 3.1.117: Determine a and b so that (2, 1) is a solution of this system: e ax...
 3.1.118: Write a system of equations having {(2, 7)} as a solution set. (Mo...
 3.1.119: Solve the system for x and y in terms of a1 , b1 , c1 , a2 , b2 , a...
 3.1.120: Solve: 6x = 10 + 5(3x  4). (Section 1.4, Example 3)
 3.1.121: Simplify: (4x2 y4 ) 2 (2x5 y0 ) 3 . (Section 1.6, Example 9)
 3.1.122: If f(x) = x2  3x + 7, find f(1). (Section 2.1, Example 3)
 3.1.123: The formula I = Pr is used to find the simple interest, I, earned f...
 3.1.124: A chemist working on a flu vaccine needs to obtain 50 milliliters o...
 3.1.125: A company that manufactures running shoes sells them at $80 per pai...
Solutions for Chapter 3.1: Systems of Linear Equations in Two Variables
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 3.1: Systems of Linear Equations in Two Variables
Get Full SolutionsChapter 3.1: Systems of Linear Equations in Two Variables includes 125 full stepbystep solutions. Since 125 problems in chapter 3.1: Systems of Linear Equations in Two Variables have been answered, more than 87990 students have viewed full stepbystep solutions from this chapter. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6.

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).

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

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.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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).

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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.