Your Algebra Homework Can Now Be Easier Than Ever!

Systems of Linear Equations

4.3. A Short Quiz to Test Yourself

Now try a few for yourselves. After you solve the problem , click on the green link
of your choice to get immediate feedback to your choice.

Quiz Solve each of the fol lowing systems using any method you choose .

1.

(4,−1)
(−9, 3)
(−8, 5)
(2, 2)
No solution
{(x, y) | x + 3y = 0}

2.

(4,−1)
(−9, 3)
(−8, 5)
(2, 2)
No Solution
{(x, y) | x − 5y = −3}

3.

(4,−1)
(−9, 3)
(−8, 5)
(2, 2)
No solution
{(x, y) | 2x − y = 9}

4.

(4,−1)
(−9, 3)
(−8, 5)
(2, 2)
No solution
{(x, y) | −x + 3y = 4}

5.

(4,−1)
(−9, 3)
(−8, 5)
(2, 2)
No solution
{(x, y) | 2x+3y = −1}

How did you do?

Solutions to Examples

Example 2.1(a): We substitute the point (−5, 3) into the equation to get

You can see that the first equation is satisfied, but the second equation is not.

Conclusion: The point (−5, 3) does not satisfy the system.

Example 2.1(b): We substitute the point (−3, 4) into the equation to get

You can see that the both equations are satisfied.

Conclusion: The point (−3, 4) does satisfy the system.

Example 4.1(a): We solve for y in the first equation of   to get
We now substitute into the second equation:

second equation
substitution
combine
and solve

Now substituting x = 6 into the we get y = −2.

Solution: These are consistent equations with a unique solution of (6,−2) , or
the solution set is { (6,−2) } .

Example 4.1(b): We solve for y in the first equation of to
get y = 1− 3x, and substitute into the second equation:

−6x − 2y = −4 second equation
−6x − 2(1 − 3x) = −4 substitution
−6x − 2 + 6x = −5 combine
−2 = −5 simplify (2)

We see that in equation (2) we get a “false” equation. This means that there is
no solution to the system.

Solution: This is an inconsistent system of equations, there is no solution to this
system. The solution set is .

Example 4.1(c): We solve for x in the first equation in to get
x = 6− 3y. Substitute into the second equation:

second equation
substitution
factor out 3
combine
deal with the minus sign , correctly
simplify again (3)

Equation (3) is typical of a dependent system of equations. The equation is always
true regardless of the value of x . Dependent systems means essentially that there
are two equations , but each equation is describing the same line .

Solution: This is an consistent system of dependent equations. Any point that
satisfies one of the equations automatically satisfies the the other equation. Therefore,
the solution set is the set of all points on the satisfy either of the two equations.
We can write { (x, y) | x + 3y = 6}

Example 4.4(a): To solve

we simply add the two equations together to get 3y = 9, hence, y = 3. From the
first equation, we see that x = −5 − x = −5 −3 = −8.

Conclusion: This is a consistent system of equations with a unique solution of
(−8, 3) .

Example 4.4(b): As discussed earlier, we multiply the first equation by −3:

given

multiply by −3
add the two equations together
solve for y

From the first equation x = −5 − 2y = −5 − 2(−8) = −5 + 16 = 11

Conclusions: This is a consistent system of equations with a unique solution of
(11,−8) .
 

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 April 25th 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 © 2004-2018, Algebra-Answer.Com.  All rights reserved.