 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 32089 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.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.