Your Algebra Homework Can Now Be Easier Than Ever!

Math 141 - Week in Review #2

Section 2.1 - Systems of Linear Equations

• When a system of linear equations has only two variables , each equation represents a line and “solving the system”
means finding all points the lines have in common .

• For any system of linear equations (with finitely many variables ), there are only 3 possibilities for the solution:
(1) a unique solution, (2) infinitely many solutions, or (3) no solution.

• If a system of equations has infinitely many solutions, you MUST give the parametric solution for the system .

Section 2.4 - Matrices

• The size of a matrix is always number of rows × number of columns.

• cij represents the entry of the matrix C in row i and column j.

• To add and subtract matrices , they must be the same size.

• When adding or subtracting matrices, add or subtract corresponding entries.

• A scalar product is computed by multiplying each entry of a matrix by a scalar (a number).

• Transpose - The rows of the matrix A become the columns of AT .

Section 2.5 - Multiplication of Matrices

• The matrix product AB can be computed only if the number of columns of A equals the number of rows of B.

• If C = AB, then cij is computed by multiplying the ith row of A by the jth column of B.

• Identity Matrix - Denoted by In, the identity matrix is the n×n matrix with 1’s down the main diagonal (from upper
left corner to lower right corner) and 0’s for all other entries.

• If A is m×n, then AIn = A and ImA = A.

Section 2.6 - The Inverse of a Square Matrix

• Only square matrices can have inverses, but not all square matrices have inverses.

• A square matrix that does not have an inverse is called a singular matrix.

• The inverse of A, denoted A-1, is the square n×n matrix such that AA-1 = A-1A = In.

• Systems of equations can be represented as a matrix equation of the form AX = B where A is the coefficient matrix ,
X is a column vector containing the variables, and B is a column vector containing constants.

• If A has an inverse, the solution to the matrix equation is X = A-1B.

• If A does not have an inverse (i.e., if A is singular), this does NOT imply the system has no solution. It simply
means that you must use another method to solve the system .

1. Solve the system of equations

2. Solve the system of equations

3x−6y = 18

−2x+4y = −12

3. (a) Find the value of k so that the given system has no solution.

7x−5y = −3

3x+ky = 15

(b) Is it possible to find a value of k so that the system has infinitely many solutions? Explain.

(c) For what value(s) of k will the system have a unique solution?

For the next 3 exercises, set up the system of equations but do not solve.

4. (#49, pg. 74 of Finite Mathematics by Lial , et. al.) The U-Drive Rent-A-Truck Co. plans to spend $6 million on 200 new vehicles. Each
van will cost $20,000, each small truck $30,000, and each large truck $50,000. Past experience shows that they
need twice as many vans as small trucks. How many of each kind of vehicle can they buy?

5. A cashier has a total of 96 bills in his register in one-, five-, and ten-dollar denominations. If he has three times as
many fives as ones, and if the number of ones and fives combined is half of the number of tens he has, how many
bills of each denomination does he have in his register?

6. Random, Inc. makes picture collates in three sizes. A small collage requires 30 minutes of cutting time and 36
minutes of pasting time. A medium collage requires 60 minutes of cutting time and 54 minutes for pasting. A
large collage requires 90 minutes for cutting and 72 minutes for pasting. There are 380 labor hours available for
cutting and 330 labor hours available for pasting each week. If the company wants to run at full capacity, how
many collages of each size should be made each week?

7. Let , and . Compute each of the

following:

(a) B+3D

(b) 2C+B

(c) 4D−3CT



(e) DB

(f) BC

(g) CDT

(h) BBT

(i) A2

8. Solve for x and y:

9. The times (in minutes) required for assembling, testing, and packaging large and small capacity food processors
are shown in the following table:

  Assembling Testing Packaging
Large
Small
45
30
15
10
10
5

(a) Define a matrix T that summarizes the above data.

(b) Let M =[100 200]represent the number of large and small food processors ordered , respectively. Find
MT and explain the meaning of its entries.

(c) If assembling costs $3 per minute, testing costs $1 per minute, and packaging costs $2 per minute, find a
matrix C that, when multiplied with T, gives the total cost for making each size of food processor.

10. If find A-1.

11. If , find B-1.

12. Solve the following system of equations using matrix inverses.

Prev Next

Start solving your Algebra Problems in next 5 minutes!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath

Attention: We are currently running a special promotional offer for Algebra-Answer.com visitors -- if you order Algebra Helper by midnight of March 29th you will pay only $39.99 instead of our regular price of $74.99 -- this is $35 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the left, not through our regular order page.

If you order now you will also receive 30 minute live session from tutor.com for a 1$!

You Will Learn Algebra Better - Guaranteed!

Just take a look how incredibly simple Algebra Helper is:

Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:

Step 2 : Let Algebra Helper solve it:

Step 3 : Ask for an explanation for the steps you don't understand:



Algebra Helper can solve problems in all the following areas:

  • simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
  • factoring and expanding expressions
  • finding LCM and GCF
  • (simplifying, rationalizing complex denominators...)
  • solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
  • solving a system of two and three linear equations (including Cramer's rule)
  • graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
  • graphing general functions
  • operations with functions (composition, inverse, range, domain...)
  • simplifying logarithms
  • basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
  • arithmetic and other pre-algebra topics (ratios, proportions, measurements...)

ORDER NOW!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath
Check out our demo!
 
"It really helped me with my homework.  I was stuck on some problems and your software walked me step by step through the process..."
C. Sievert, KY
 
 
Sofmath
19179 Blanco #105-234
San Antonio, TX 78258
Phone: (512) 788-5675
Fax: (512) 519-1805
 

Home   : :   Features   : :   Demo   : :   FAQ   : :   Order

Copyright © 2004-2024, Algebra-Answer.Com.  All rights reserved.