 3.3.1: Use the indicated points to find the slope of each line. See Exampl...
 3.3.2: Use the indicated points to find the slope of each line. See Exampl...
 3.3.3: Use the indicated points to find the slope of each line. See Exampl...
 3.3.4: Use the indicated points to find the slope of each line. See Exampl...
 3.3.5: Use the indicated points to find the slope of each line. See Exampl...
 3.3.6: Use the indicated points to find the slope of each line. See Exampl...
 3.3.7: In the context of the graph of a straight line, what is meant by ri...
 3.3.8: Explain in your own words what is meant by slope of a line.
 3.3.9: Match the graph of each line in (a)(d) with its slope in AD.(a) (b)...
 3.3.10: Decide whether the line with the given slope rises from left to rig...
 3.3.11: On a pair of axes similar to the one shown, sketch the graph of a s...
 3.3.12: On a pair of axes similar to the one shown, sketch the graph of a s...
 3.3.13: On a pair of axes similar to the one shown, sketch the graph of a s...
 3.3.14: On a pair of axes similar to the one shown, sketch the graph of a s...
 3.3.15: The figure at the right shows a line that has a positive slope (bec...
 3.3.16: The figure at the right shows a line that has a positive slope (bec...
 3.3.17: The figure at the right shows a line that has a positive slope (bec...
 3.3.18: The figure at the right shows a line that has a positive slope (bec...
 3.3.19: The figure at the right shows a line that has a positive slope (bec...
 3.3.20: The figure at the right shows a line that has a positive slope (bec...
 3.3.21: What is the slope (or grade) of this hill?
 3.3.22: What is the slope (or pitch) of this roof ?
 3.3.23: What is the slope of the slide? (Hint: The slide drops 8 ft vertica...
 3.3.24: What is the slope (or grade) of this ski slope? (Hint: The ski slop...
 3.3.25: A student was asked to find the slope of the line through the point...
 3.3.26: A student was asked to find the slope of the line through the point...
 3.3.27: Find the slope of the line through each pair of points. See Example...
 3.3.28: Find the slope of the line through each pair of points. See Example...
 3.3.29: Find the slope of the line through each pair of points. See Example...
 3.3.30: Find the slope of the line through each pair of points. See Example...
 3.3.31: Find the slope of the line through each pair of points. See Example...
 3.3.32: Find the slope of the line through each pair of points. See Example...
 3.3.33: Find the slope of the line through each pair of points. See Example...
 3.3.34: Find the slope of the line through each pair of points. See Example...
 3.3.35: Find the slope of the line through each pair of points. See Example...
 3.3.36: Find the slope of the line through each pair of points. See Example...
 3.3.37: Find the slope of the line through each pair of points. See Example...
 3.3.38: Find the slope of the line through each pair of points. See Example...
 3.3.39: Find the slope of the line through each pair of points. See Example...
 3.3.40: Find the slope of the line through each pair of points. See Example...
 3.3.41: Find the slope of each line. See Example 5y = 5x + 12
 3.3.42: Find the slope of each line. See Example 5y = 2x + 3
 3.3.43: Find the slope of each line. See Example 54y = x + 1
 3.3.44: Find the slope of each line. See Example 52y = x + 4
 3.3.45: Find the slope of each line. See Example 53x  2y = 3
 3.3.46: Find the slope of each line. See Example 56x  4y = 4
 3.3.47: Find the slope of each line. See Example 53x + 2y = 5
 3.3.48: Find the slope of each line. See Example 52x + 4y = 5
 3.3.49: Find the slope of each line. See Example 5y = 5
 3.3.50: Find the slope of each line. See Example 5y = 4
 3.3.51: Find the slope of each line. See Example 5x = 6
 3.3.52: Find the slope of each line. See Example 5x = 2
 3.3.53: What is the slope of a line whose graph is parallel to the graph of...
 3.3.54: What is the slope of a line whose graph is parallel to the graph of...
 3.3.55: If two lines are both vertical or both horizontal, which of the fol...
 3.3.56: If a line is vertical, what is true of any line that is perpendicul...
 3.3.57: For each pair of equations, give the slopes of the lines and then d...
 3.3.58: For each pair of equations, give the slopes of the lines and then d...
 3.3.59: For each pair of equations, give the slopes of the lines and then d...
 3.3.60: For each pair of equations, give the slopes of the lines and then d...
 3.3.61: For each pair of equations, give the slopes of the lines and then d...
 3.3.62: For each pair of equations, give the slopes of the lines and then d...
 3.3.63: For each pair of equations, give the slopes of the lines and then d...
 3.3.64: For each pair of equations, give the slopes of the lines and then d...
 3.3.65: Work Exercises 6570 in orderUse the ordered pairs 1990, 11,338 and ...
 3.3.66: Work Exercises 6570 in orderThe slope of the line in FIGURE A is . ...
 3.3.67: Work Exercises 6570 in orderThe slope of a line represents the rate...
 3.3.68: Work Exercises 6570 in orderUse the given information to find the s...
 3.3.69: Work Exercises 6570 in orderThe slope of the line in FIGURE B is . ...
 3.3.70: Work Exercises 6570 in orderOn the basis of FIGURE B, what was the ...
 3.3.71: The graph shows album sales (which include CD, vinyl, cassette, and...
 3.3.72: The graph shows album sales (which include CD, vinyl, cassette, and...
 3.3.73: Some graphing calculators have the capability of displaying a table...
 3.3.74: Some graphing calculators have the capability of displaying a table...
 3.3.75: Some graphing calculators have the capability of displaying a table...
 3.3.76: Some graphing calculators have the capability of displaying a table...
 3.3.77: Solve each equation for y. See Section 2.5.2x + 5y = 15
 3.3.78: Solve each equation for y. See Section 2.5.4x + 3y = 8
 3.3.79: Solve each equation for y. See Section 2.5.10x = 30 + 3y
 3.3.80: Solve each equation for y. See Section 2.5.8x = 8  2y
 3.3.81: Solve each equation for y. See Section 2.5.y  182 = 21x  42
 3.3.82: Solve each equation for y. See Section 2.5.y  3 = 43x  1624
Solutions for Chapter 3.3: The Slope of a Line
Full solutions for Beginning Algebra  11th Edition
ISBN: 9780321673480
Solutions for Chapter 3.3: The Slope of a Line
Get Full SolutionsThis textbook survival guide was created for the textbook: Beginning Algebra, edition: 11. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.3: The Slope of a Line includes 82 full stepbystep solutions. Beginning Algebra was written by and is associated to the ISBN: 9780321673480. Since 82 problems in chapter 3.3: The Slope of a Line have been answered, more than 82639 students have viewed full stepbystep solutions from this chapter.

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).

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

GramSchmidt 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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

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.

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

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.

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

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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.