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Solutions for Chapter 4-4: Writing Equations in Slope-Intercept Form

Algebra 1, Student Edition (MERRILL ALGEBRA 1) | 1st Edition | ISBN: 9780078738227 | Authors: Berchie Holliday, Gilbert J. Cuevas, Beatrice Luchin, Ruth M. Casey, Linda M. Hayek, John A. Carter, Daniel Marks, Roger Day, & 2 more

Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1) | 1st Edition

ISBN: 9780078738227

Algebra 1, Student Edition (MERRILL ALGEBRA 1) | 1st Edition | ISBN: 9780078738227 | Authors: Berchie Holliday, Gilbert J. Cuevas, Beatrice Luchin, Ruth M. Casey, Linda M. Hayek, John A. Carter, Daniel Marks, Roger Day, & 2 more

Solutions for Chapter 4-4: Writing Equations in Slope-Intercept Form

Solutions for Chapter 4-4
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Textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1)
Edition: 1
Author: Berchie Holliday, Gilbert J. Cuevas, Beatrice Luchin, Ruth M. Casey, Linda M. Hayek, John A. Carter, Daniel Marks, Roger Day, & 2 more
ISBN: 9780078738227

Chapter 4-4: Writing Equations in Slope-Intercept Form includes 53 full step-by-step solutions. Since 53 problems in chapter 4-4: Writing Equations in Slope-Intercept Form have been answered, more than 35277 students have viewed full step-by-step solutions from this chapter. Algebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1) , edition: 1.

Key Math Terms and definitions covered in this textbook
  • Back substitution.

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

  • Change of basis matrix M.

    The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

  • Complete solution x = x p + Xn to Ax = b.

    (Particular x p) + (x n in nullspace).

  • Covariance matrix:E.

    When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

  • Distributive Law

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

  • Fibonacci numbers

    0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

  • Gram-Schmidt orthogonalization A = QR.

    Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

  • Hessenberg matrix H.

    Triangular matrix with one extra nonzero adjacent diagonal.

  • Least squares solution X.

    The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

  • Left inverse A+.

    If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

  • Left nullspace N (AT).

    Nullspace of AT = "left nullspace" of A because y T A = OT.

  • Minimal polynomial of A.

    The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

  • Multiplication Ax

    = Xl (column 1) + ... + xn(column n) = combination of columns.

  • Multiplicities AM and G M.

    The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

  • Plane (or hyperplane) in Rn.

    Vectors x with aT x = O. Plane is perpendicular to a =1= O.

  • Right inverse A+.

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

  • Simplex method for linear programming.

    The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

  • Singular matrix A.

    A square matrix that has no inverse: det(A) = o.

  • Spectral Theorem A = QAQT.

    Real symmetric A has real A'S and orthonormal q's.

  • Symmetric matrix A.

    The transpose is AT = A, and aU = a ji. A-I is also symmetric.

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