 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 24572 students have viewed full stepbystep solutions from this chapter.

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

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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.

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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

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

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.