 4.2.1: A student solves the following system and finds that , which is cor...
 4.2.2: A student solves the following system and obtains the equation . Th...
 4.2.3: Solve each system by the substitution method. Check each solution. ...
 4.2.4: Solve each system by the substitution method. Check each solution. ...
 4.2.5: Solve each system by the substitution method. Check each solution. ...
 4.2.6: Solve each system by the substitution method. Check each solution. ...
 4.2.7: Solve each system by the substitution method. Check each solution. ...
 4.2.8: Solve each system by the substitution method. Check each solution. ...
 4.2.9: Solve each system by the substitution method. Check each solution. ...
 4.2.10: Solve each system by the substitution method. Check each solution. ...
 4.2.11: Solve each system by the substitution method. Check each solution. ...
 4.2.12: Solve each system by the substitution method. Check each solution. ...
 4.2.13: Solve each system by the substitution method. Check each solution. ...
 4.2.14: Solve each system by the substitution method. Check each solution. ...
 4.2.15: Solve each system by the substitution method. Check each solution. ...
 4.2.16: Solve each system by the substitution method. Check each solution. ...
 4.2.17: Solve each system by the substitution method. Check each solution. ...
 4.2.18: Solve each system by the substitution method. Check each solution. ...
 4.2.19: Solve each system by the substitution method. Check each solution. ...
 4.2.20: Solve each system by the substitution method. Check each solution. ...
 4.2.21: Solve each system by the substitution method. Check each solution. ...
 4.2.22: Solve each system by the substitution method. Check each solution. ...
 4.2.23: Solve each system by the substitution method. Check each solution. ...
 4.2.24: Solve each system by the substitution method. Check each solution. ...
 4.2.25: Solve each system by the substitution method. Check each solution. ...
 4.2.26: Solve each system by the substitution method. Check each solution. ...
 4.2.27: Solve each system by the substitution method. Check each solution. ...
 4.2.28: Solve each system by the substitution method. Check each solution. ...
 4.2.29: Solve each system by the substitution method. Check each solution. ...
 4.2.30: Solve each system by the substitution method. Check each solution. ...
 4.2.31: Solve each system by the substitution method. Check each solution. ...
 4.2.32: Solve each system by the substitution method. Check each solution. ...
 4.2.33: A system of linear equations can be used to model the cost and the ...
 4.2.34: A system of linear equations can be used to model the cost and the ...
 4.2.35: A system of linear equations can be used to model the cost and the ...
 4.2.36: A system of linear equations can be used to model the cost and the ...
 4.2.37: Solve each system by substitution. Then graph both lines in the sta...
 4.2.38: Solve each system by substitution. Then graph both lines in the sta...
 4.2.39: Solve each system by substitution. Then graph both lines in the sta...
 4.2.40: Solve each system by substitution. Then graph both lines in the sta...
 4.2.41: Solve each system by substitution. Then graph both lines in the sta...
 4.2.42: Solve each system by substitution. Then graph both lines in the sta...
 4.2.43: Simplify. See Section 1.8114x  3y2 + 12x + 3y2
 4.2.44: Simplify. See Section 1.816x + 8y2 + 16x + 2y2
 4.2.45: Simplify. See Section 1.81x + 7y2 + 13y + x
 4.2.46: Simplify. See Section 1.83x  4y2 + 14y  3x2
 4.2.47: What must be added to to get a sum of 0?
 4.2.48: What must be added to 6y to get a sum of 0?
 4.2.49: What must 4y be multiplied by so that when the product is added to ...
 4.2.50: What must be multiplied by so that when the product is added to x, ...
Solutions for Chapter 4.2: Solving Systems of Linear Equations by Substitution
Full solutions for Beginning Algebra  11th Edition
ISBN: 9780321673480
Solutions for Chapter 4.2: Solving Systems of Linear Equations by Substitution
Get Full SolutionsThis textbook survival guide was created for the textbook: Beginning Algebra, edition: 11. Beginning Algebra was written by and is associated to the ISBN: 9780321673480. Chapter 4.2: Solving Systems of Linear Equations by Substitution includes 50 full stepbystep solutions. Since 50 problems in chapter 4.2: Solving Systems of Linear Equations by Substitution have been answered, more than 37952 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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).

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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.)

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

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.

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.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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·

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.