 2.1.2.1.1: Solve each equation and check. See Examples 1 and 2.5x = 30
 2.1.2.1.2: Solve each equation and check. See Examples 1 and 2.2x = 18
 2.1.2.1.3: Solve each equation and check. See Examples 1 and 2.10 = x + 12
 2.1.2.1.4: Solve each equation and check. See Examples 1 and 2.25 = y + 30
 2.1.2.1.5: Solve each equation and check. See Examples 1 and 2.x  2.8 = 1.9
 2.1.2.1.6: Solve each equation and check. See Examples 1 and 2.y  8.6 = 6.3
 2.1.2.1.7: Solve each equation and check. See Examples 1 and 2.5x  4 = 26 + 2x
 2.1.2.1.8: Solve each equation and check. See Examples 1 and 2.5y  3 = 11 + 3
 2.1.2.1.9: Solve each equation and check. See Examples 1 and 2.4.1  7z = 3.6
 2.1.2.1.10: Solve each equation and check. See Examples 1 and 2.10.3  6x = 2.3
 2.1.2.1.11: Solve each equation and check. See Examples 1 and 2.5y + 12 = 2y  3
 2.1.2.1.12: Solve each equation and check. See Examples 1 and 2.4x + 14 = 6x + 8
 2.1.2.1.13: Solve each equation and check. See Examples 3 and 43x  4  5x = x ...
 2.1.2.1.14: Solve each equation and check. See Examples 3 and 413x  15x + 8 = ...
 2.1.2.1.15: Solve each equation and check. See Examples 3 and 48x  5x + 3 = x ...
 2.1.2.1.16: Solve each equation and check. See Examples 3 and 46 + 3x + x = x ...
 2.1.2.1.17: Solve each equation and check. See Examples 3 and 45x + 12 = 212x + 72
 2.1.2.1.18: Solve each equation and check. See Examples 3 and 4214x + 32 = 7x + 5
 2.1.2.1.19: Solve each equation and check. See Examples 3 and 431x  62 = 5x
 2.1.2.1.20: Solve each equation and check. See Examples 3 and 46x = 41x  52
 2.1.2.1.21: Solve each equation and check. See Examples 3 and 4215y  12  y =...
 2.1.2.1.22: Solve each equation and check. See Examples 3 and 4413n  22  n =...
 2.1.2.1.23: Solve each equation and check. See Examples 5 through 7.x2+ x3 = 34
 2.1.2.1.24: Solve each equation and check. See Examples 5 through 7.x2+ x5 = 54
 2.1.2.1.25: Solve each equation and check. See Examples 5 through 7. 3t4  t2 = 1
 2.1.2.1.26: Solve each equation and check. See Examples 5 through 7.4r5  r10 = 7
 2.1.2.1.27: Solve each equation and check. See Examples 5 through 7.n  34+ n +...
 2.1.2.1.28: Solve each equation and check. See Examples 5 through 7.2 + h9+h  ...
 2.1.2.1.29: Solve each equation and check. See Examples 5 through 7.0.6x  10 =...
 2.1.2.1.30: Solve each equation and check. See Examples 5 through 7.0.3x + 2.4 ...
 2.1.2.1.31: Solve each equation and check. See Examples 5 through 7.3x  19+ x ...
 2.1.2.1.32: Solve each equation and check. See Examples 5 through 7.2z + 78  2...
 2.1.2.1.33: Solve each equation and check. See Examples 5 through 7.1.514  x2 ...
 2.1.2.1.34: Solve each equation and check. See Examples 5 through 7.2.412x + 32...
 2.1.2.1.35: Solve each equation. See Examples 8 and 9.41n + 32 = 216 + 2n2
 2.1.2.1.36: Solve each equation. See Examples 8 and 9.614n + 42 = 813 + 3n2
 2.1.2.1.37: Solve each equation. See Examples 8 and 9.31x + 12 + 5 = 3x + 2
 2.1.2.1.38: Solve each equation. See Examples 8 and 9.41x + 22 + 4 = 4x  8
 2.1.2.1.39: Solve each equation. See Examples 8 and 9.21x  82 + x = 31x  62 + 2
 2.1.2.1.40: Solve each equation. See Examples 8 and 9.51x  42 + x = 61x  22  8
 2.1.2.1.41: Solve each equation. See Examples 8 and 9.41x + 52 = 31x  42 + x
 2.1.2.1.42: Solve each equation. See Examples 8 and 9.91x  22 = 81x  32 + x
 2.1.2.1.43: Solve each equation. See Examples 1 through 9.38+b3 = 512
 2.1.2.1.44: Solve each equation. See Examples 1 through 9.a2+74 = 5
 2.1.2.1.45: Solve each equation. See Examples 1 through 9.x  10 = 6x  10
 2.1.2.1.46: Solve each equation. See Examples 1 through 9.4x  7 = 2x  7
 2.1.2.1.47: Solve each equation. See Examples 1 through 9.51x  22 + 2x = 71x +...
 2.1.2.1.48: Solve each equation. See Examples 1 through 9.3x + 21x + 42 = 51x +...
 2.1.2.1.49: Solve each equation. See Examples 1 through 9.y + 0.2 = 0.61y + 32
 2.1.2.1.50: Solve each equation. See Examples 1 through 9.1w + 0.22 = 0.314  w2
 2.1.2.1.51: Solve each equation. See Examples 1 through 9.141a + 22 = 1615  a2
 2.1.2.1.52: Solve each equation. See Examples 1 through 9.1318 + 2c2 = 1513c  52
 2.1.2.1.53: Solve each equation. See Examples 1 through 9.2y + 51y  42 = 4y  ...
 2.1.2.1.54: Solve each equation. See Examples 1 through 9.9c  316  5c2 = c  ...
 2.1.2.1.55: Solve each equation. See Examples 1 through 9.6x  21x  32 = 41x +...
 2.1.2.1.56: Solve each equation. See Examples 1 through 9.10x  21x + 42 = 81x ...
 2.1.2.1.57: Solve each equation. See Examples 1 through 9.m  43  3m  15 = 1
 2.1.2.1.58: Solve each equation. See Examples 1 through 9.n + 18  2  n3 = 56
 2.1.2.1.59: Solve each equation. See Examples 1 through 9.8x  12  3x = 9x  7
 2.1.2.1.60: Solve each equation. See Examples 1 through 9.10y  18  4y = 12y  13
 2.1.2.1.61: Solve each equation. See Examples 1 through 9.13x  52  12x  62 ...
 2.1.2.1.62: Solve each equation. See Examples 1 through 9.412x  32  110x + 7...
 2.1.2.1.63: Solve each equation. See Examples 1 through 9. 131y + 42 + 6 = 1413...
 2.1.2.1.64: Solve each equation. See Examples 1 through 9.1512y  12  2 = 1213...
 2.1.2.1.65: Solve each equation. See Examples 1 through 9.237  511  n24 + 8n ...
 2.1.2.1.66: Solve each equation. See Examples 1 through 9.338  41n  224 + 5n ...
 2.1.2.1.67: Translating. Translate each phrase into an expression. Use the vari...
 2.1.2.1.68: Translating. Translate each phrase into an expression. Use the vari...
 2.1.2.1.69: Translating. Translate each phrase into an expression. Use the vari...
 2.1.2.1.70: Translating. Translate each phrase into an expression. Use the vari...
 2.1.2.1.71: Translating. Translate each phrase into an expression. Use the vari...
 2.1.2.1.72: Translating. Translate each phrase into an expression. Use the vari...
 2.1.2.1.73: Find the error for each proposed solution. Then correct the propose...
 2.1.2.1.74: Find the error for each proposed solution. Then correct the propose...
 2.1.2.1.75: Find the error for each proposed solution. Then correct the propose...
 2.1.2.1.76: Find the error for each proposed solution. Then correct the propose...
 2.1.2.1.77: By inspection, decide which equations have no solution and which eq...
 2.1.2.1.78: By inspection, decide which equations have no solution and which eq...
 2.1.2.1.79: By inspection, decide which equations have no solution and which eq...
 2.1.2.1.80: By inspection, decide which equations have no solution and which eq...
 2.1.2.1.81: a. Simplify the expression 41x + 12 + 1.b. Solve the equation 41x +...
 2.1.2.1.82: Explain why the multiplication property of equality does not includ...
 2.1.2.1.83: In your own words, explain why the equation x + 7 = x + 6 has no so...
 2.1.2.1.84: In your own words, explain why the equation x = x has one solution...
 2.1.2.1.85: Find the value of K such that the equations are equivalent.3.2x + 4...
 2.1.2.1.86: Find the value of K such that the equations are equivalent.7.6y  ...
 2.1.2.1.87: Find the value of K such that the equations are equivalent.711x + 9...
 2.1.2.1.88: Find the value of K such that the equations are equivalent.x6+ 4 = ...
 2.1.2.1.89: Write a linear equation in x whose only solution is 5.
 2.1.2.1.90: Write an equation in x that has no solution.
 2.1.2.1.91: Solve the following.x1x  62 + 7 = x1x + 12
 2.1.2.1.92: Solve the following.7x2 + 2x  3 = 6x1x + 42 + x2
 2.1.2.1.93: Solve the following.3x1x + 52  12 = 3x2 + 10x + 3
 2.1.2.1.94: Solve the following.x1x + 12 + 16 = x1x + 52
 2.1.2.1.95: Solve and check.2.569x = 12.48534
 2.1.2.1.96: Solve and check.9.112y = 47.537304
 2.1.2.1.97: Solve and check.2.86z  8.1258 = 3.75
 2.1.2.1.98: Solve and check.1.25x  20.175 = 8.15
Solutions for Chapter 2.1: Linear Equations in One Variable
Full solutions for Intermediate Algebra  6th Edition
ISBN: 9780321785046
Solutions for Chapter 2.1: Linear Equations in One Variable
Get Full SolutionsThis textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6. Chapter 2.1: Linear Equations in One Variable includes 98 full stepbystep solutions. Since 98 problems in chapter 2.1: Linear Equations in One Variable have been answered, more than 65538 students have viewed full stepbystep solutions from this chapter. Intermediate Algebra was written by and is associated to the ISBN: 9780321785046. This expansive textbook survival guide covers the following chapters and their solutions.

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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.

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

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.