 7.2.7.2.1: The first step in solving a system of equations by the _______ of _...
 7.2.7.2.2: Two systems of equations that have the same solution set are called...
 7.2.7.2.3: A system of linear equations that has at least one solution is call...
 7.2.7.2.4: In Exercises 16, solve the system by the method of elimination. Lab...
 7.2.7.2.5: In Exercises 16, solve the system by the method of elimination. Lab...
 7.2.7.2.6: In Exercises 16, solve the system by the method of elimination. Lab...
 7.2.7.2.7: In Exercises 716, solve the system by the method of elimination and...
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 7.2.7.2.16: In Exercises 716, solve the system by the method of elimination and...
 7.2.7.2.17: In Exercises 1720, match the system of linear equations with its gr...
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 7.2.7.2.21: In Exercises 2140, solve the system by the method of elimination an...
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 7.2.7.2.40: In Exercises 2140, solve the system by the method of elimination an...
 7.2.7.2.41: In Exercises 41 46, use a graphing utility to graph the lines in th...
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 7.2.7.2.47: In Exercises 4754, use a graphing utility to graph the two equation...
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 7.2.7.2.55: In Exercises 5562, use any method to solve the system.
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 7.2.7.2.61: In Exercises 5562, use any method to solve the system.
 7.2.7.2.62: In Exercises 5562, use any method to solve the system.
 7.2.7.2.63: Exploration In Exercises 6366, find a system of linear equations th...
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 7.2.7.2.65: Exploration In Exercises 6366, find a system of linear equations th...
 7.2.7.2.66: Exploration In Exercises 6366, find a system of linear equations th...
 7.2.7.2.67: Supply and Demand In Exercises 6770, find the point of equilibrium ...
 7.2.7.2.68: Supply and Demand In Exercises 6770, find the point of equilibrium ...
 7.2.7.2.69: Supply and Demand In Exercises 6770, find the point of equilibrium ...
 7.2.7.2.70: Supply and Demand In Exercises 6770, find the point of equilibrium ...
 7.2.7.2.71: Airplane Speed An airplane flying into a headwind travels the 1800...
 7.2.7.2.72: Airplane Speed Two planes start from Bostons Logan International Ai...
 7.2.7.2.73: Ticket Sales A minor league baseball team had a total attendance on...
 7.2.7.2.74: Consumerism One family purchases five cold drinks and three snow co...
 7.2.7.2.75: Produce A grocer sells oranges for $0.95 each and grapefruits for $...
 7.2.7.2.76: Sales The sales S (in millions of dollars) for Family Dollar Stores...
 7.2.7.2.77: Revenues Revenues for a video rental store on a particular Friday e...
 7.2.7.2.78: Sales On Saturday night, the manager of a shoe store evaluates the ...
 7.2.7.2.79: Fitting a Line to Data In Exercises 7982, find the least squares re...
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 7.2.7.2.83: Data Analysis A farmer used four test plots to determine the relati...
 7.2.7.2.84: Data Analysis A candy store manager wants to know the demand for a ...
 7.2.7.2.85: True or False? In Exercises 85 and 86, determine whether the statem...
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 7.2.7.2.87: Think About It In Exercises 87 and 88, the graphs of the two equati...
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 7.2.7.2.89: Writing Briefly explain whether or not it is possible for a consist...
 7.2.7.2.90: Think About It Give examples of (a) a system of linear equations th...
 7.2.7.2.91: In Exercises 91 and 92, find the value of k such that the system of...
 7.2.7.2.92: In Exercises 91 and 92, find the value of k such that the system of...
 7.2.7.2.93: Advanced Applications In Exercises 93 and 94, solve the system of e...
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 7.2.7.2.95: In Exercises 95100, solve the inequality and graph the solution on ...
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 7.2.7.2.101: In Exercises 101106, write the expression as the logarithm of a sin...
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 7.2.7.2.106: In Exercises 101106, write the expression as the logarithm of a sin...
 7.2.7.2.107: Make a Decision To work an extended application analyzing the avera...
Solutions for Chapter 7.2: Systems of Linear Equations in Two Variables
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 7.2: Systems of Linear Equations in Two Variables
Get Full SolutionsThis textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Since 107 problems in chapter 7.2: Systems of Linear Equations in Two Variables have been answered, more than 44725 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 7.2: Systems of Linear Equations in Two Variables includes 107 full stepbystep solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522.

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

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.

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det 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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

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.

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.

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

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·