 3.3.1: A solution of a system of linear equations in three variables is an...
 3.3.2: Consider the following system: x + y  z = 1 2x  2y  5z = 7 4x +...
 3.3.3: Consider the following system: x + y + z = 2 2x  3y = 3 10y  z = ...
 3.3.4: A function of the form y = ax2 + bx + c, a 0, is called a/an function.
 3.3.5: The process of determining a function whose graph contains given po...
 3.3.6: 2x + y  2z = 1 3x  3y  z = 5 x  2y + 3z = 6
 3.3.7: 4x  y + 2z = 11 x + 2y  z = 1 2x + 2y  3z = 1
 3.3.8: x  y + 3z = 8 3x + y  2z = 2 2x + 4y + z = 0
 3.3.9: 3x + 2y  3z = 2 2x  5y + 2z = 2 4x  3y + 4z = 10
 3.3.10: 2x + 3y + 7z = 13 3x + 2y  5z = 22 5x + 7y  3z = 28
 3.3.11: 2x  4y + 3z = 17 x + 2y  z = 0 4x  y  z = 6
 3.3.12: x + z = 3 x + 2y  z = 1 2x  y + z = 3
 3.3.13: 2x + y = 2 x + y  z = 4 3x + 2y + z = 0
 3.3.14: x + 3y + 5z = 20 y  4z = 16 3x  2y + 9z = 36
 3.3.15: x + y = 4 y  z = 1 2x + y + 3z = 21
 3.3.16: x + y = 4 x + z = 4 y + z = 4
 3.3.17: 2x + y + 2z = 1 3x  y + z = 2 x  2y  z = 0
 3.3.18: 3x + 4y + 5z = 8 x  2y + 3z = 6 2x  4y + 6z = 8
 3.3.19: 5x  2y  5z = 1 10x  4y  10z = 2 15x  6y  15z = 3
 3.3.20: x + 2y + z = 4 3x  4y + z = 4 6x  8y + 2z = 8
 3.3.21: 3(2x + y) + 5z = 1 2(x  3y + 4z) = 9 4(1 + x) = 3(z  3y)
 3.3.22: 7z  3 = 2(x  3y) 5y + 3z  7 = 4x 4 + 5z = 3(2x  y)
 3.3.23: (1, 6), (1, 4), (2, 9)
 3.3.24: (2, 7), (1, 2), (2, 3)
 3.3.25: (1, 4), (1, 2), (2, 5)
 3.3.26: (1, 3), (3, 1), (4, 0)
 3.3.27: The sum of three numbers is 16. The sum of twice the first number, ...
 3.3.28: The following is known about three numbers: Three times the first n...
 3.3.29: f x + 2 6  y + 4 3 + z 2 = 0 x + 1 2 + y  1 2  z 4 = 9 2 x  5 4...
 3.3.30: f x + 3 2  y  1 2 + z + 2 4 = 3 2 x  5 2 + y + 1 3  z 4 =  25 ...
 3.3.31: In Exercises 3132, find the equation of the quadratic function y = ...
 3.3.32: In Exercises 3132, find the equation of the quadratic function y = ...
 3.3.33: ax  by  2cz = 21 ax + by + cz = 0 2ax  by + cz = 14
 3.3.34: ax  by + 2cz = 4 ax + 3by  cz = 1 2ax + by + 3cz = 2
 3.3.35: The bar graph shows the percentage of U.S. parents willing to pay f...
 3.3.36: How much time do you spend on hygiene/grooming in the morning (incl...
 3.3.37: You throw a ball straight up from a rooftop. The ball misses the ro...
 3.3.38: A mathematical model can be used to describe the relationship betwe...
 3.3.39: In this exercise, we refer to annual spending per person in 2010. T...
 3.3.40: In this exercise, we refer to annual spending per person in 1980. T...
 3.3.41: A person invested $6700 for one year, part at 8%, part at 10%, and ...
 3.3.42: A person invested $17,000 for one year, part at 10%, part at 12%, a...
 3.3.43: At a college production of Streetcar Named Desire, 400 tickets were...
 3.3.44: A certain brand of razor blades comes in packages of 6, 12, and 24 ...
 3.3.45: Three foods have the following nutritional content per ounce. Calor...
 3.3.46: A furniture company produces three types of desks: a childrens mode...
 3.3.47: What is a system of linear equations in three variables?
 3.3.48: How do you determine whether a given ordered triple is a solution o...
 3.3.49: Describe in general terms how to solve a system in three variables.
 3.3.50: Describe what happens when using algebraic methods to solve an inco...
 3.3.51: Describe what happens when using algebraic methods to solve a syste...
 3.3.52: AIDS is taking a deadly toll on southern Africa. Describe how to us...
 3.3.53: Does your graphing utility have a feature that allows you to solve ...
 3.3.54: Verify your results in Exercises 2326 by using a graphing utility t...
 3.3.55: Solving a system in three variables, I found that x = 3 and y = 1....
 3.3.56: A system of linear equations in three variables, x, y, and z, canno...
 3.3.57: Im solving a threevariable system in which one of the given equati...
 3.3.58: Because the percentage of the U.S. population that was foreignborn...
 3.3.59: The ordered triple (2, 15, 14) is the only solution of the equation...
 3.3.60: The equation x  y  z = 6 is satisfied by (2, 3, 5).
 3.3.61: If two equations in a system are x + y  z = 5 and x + y  z = 6, t...
 3.3.62: An equation with four variables, such as x + 2y  3z + 5w = 2, cann...
 3.3.63: In the following triangle, the degree measures of the three interio...
 3.3.64: A modernistic painting consists of triangles, rectangles, and penta...
 3.3.65: Two blocks of wood having the same length and width are placed on t...
 3.3.66: f(x) =  3 4 x + 3 (Section 2.4, Example 5)
 3.3.67: 2x + y = 6 (Section 2.4, Example 1)
 3.3.68: f(x) = 5 (Section 2.4, Example 6)
 3.3.69: Solve the system: e x + 2y = 1 y = 1. What makes it fairly easy to...
 3.3.70: Solve the system: x + y + 2z = 19 y + 2z = 13 z = 5. What makes it ...
 3.3.71: Consider the following array of numbers: c 1 2 1 4 3 15 d . Rewr...
Solutions for Chapter 3.3: Systems of Linear Equations in Three Variables
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 3.3: Systems of Linear Equations in Three Variables
Get Full SolutionsIntermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. This expansive textbook survival guide covers the following chapters and their solutions. Since 71 problems in chapter 3.3: Systems of Linear Equations in Three Variables have been answered, more than 6772 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6. Chapter 3.3: Systems of Linear Equations in Three Variables includes 71 full stepbystep solutions.

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

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

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

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

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

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

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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

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

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

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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!

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

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.
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