 7.1.7.1.1: A set of two or more equations in two or more unknowns is called a ...
 7.1.7.1.2: A _______ of a system of equations is an ordered pair that satisfie...
 7.1.7.1.3: The first step in solving a system of equations by the _______ of _...
 7.1.7.1.4: Graphically, the solution to a system of equations is called the __...
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 7.1.7.1.6: In Exercises 512, solve the system by the method of substitution. C...
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 7.1.7.1.29: In Exercises 2936, solve the system graphically. Verify your soluti...
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 7.1.7.1.37: In Exercises 3750, use a graphing utility to approximate all points...
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 7.1.7.1.51: In Exercises 5164, solve the system graphically or algebraically. E...
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 7.1.7.1.69: DVD Rentals The daily DVD rentals of a newly released animated film...
 7.1.7.1.70: Sports The points scored during each of the first 12 games by two p...
 7.1.7.1.71: BreakEven Analysis A small software company invests $16,000 to pro...
 7.1.7.1.72: BreakEven Analysis A small fast food restaurant invests $5000 to p...
 7.1.7.1.73: Choice of Two Jobs You are offered two different jobs selling denta...
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 7.1.7.1.75: Investment A total of $20,000 is invested in two funds paying 6.5% ...
 7.1.7.1.76: Log Volume You are offered two different rules for estimating the n...
 7.1.7.1.77: Population The populations (in thousands) of Missouri M and Tenness...
 7.1.7.1.78: Tuition The table shows the average costs (in dollars) of one years...
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 7.1.7.1.85: Think About It When solving a system of equations by substitution, ...
 7.1.7.1.86: Writing Write a brief paragraph describing any advantages of substi...
 7.1.7.1.87: Exploration Find the equations of lines whose graphs intersect the ...
 7.1.7.1.88: Exploration Create systems of two linear equations in two variables...
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 7.1.7.1.90: Conjecture Consider the system of equations. (a) Use a graphing uti...
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Solutions for Chapter 7.1: Solving Systems of Equations
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 7.1: Solving Systems of Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Chapter 7.1: Solving Systems of Equations includes 102 full stepbystep solutions. Since 102 problems in chapter 7.1: Solving Systems of Equations have been answered, more than 47445 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522.

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.

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

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.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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.

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.

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

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.

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 ().

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.