Transcript Chapter 5 sec5_1-5_5

College Algebra K/DC
Tuesday, 21 April 2015
• OBJECTIVE TSW solve 2 x 2 systems of equations
by the methods of (1) substitution and (2) elimination.
• COOKOUT: Monday, 18 May 2015
– If you will be attending, bring $3.00 by Wednesday,
13 May 2015.
– Let me know if you have any dietary restrictions.
Turn in purple
Awards Forms to
Mrs. Yerkes ASAP!!!
1
5.1 Systems of Linear Equations
Linear Systems ▪ Substitution Method ▪ Elimination Method ▪
Special Systems
5-2
Linear System of Equations
A set of first-degree equations in n unknowns is called a
linear system of equations.
The solutions of a system of equations must satisfy (make
Know these
true) every equation in the system.
vocabulary
descriptions!
There are three possibilities for the solutions of a linear
system of two equations and two variables:
1. One solution
2. No solutions
3. Infinitely many solutions
(consistent,
(inconsistent system) (consistent,
independent system)
dependent system)
5-3
Algebraic Methods to Solve Systems
Substitution Method.
•
Use one of the equations to find an expression
for one variable in terms of the other
Elimination Method.
•
Use multiplication and/or addition to eliminate a
variable from one equation.
5-4
Solving a System by Substitution
Solve the system.
Solve equation (2) for x: x = 1 + 2y
Replace x in equation (1) with 1 + 2y, then solve for y:
Distributive property
Replace y in equation (2) with 2, then solve for x:
5-5
Solving a System by Substitution
The solution of the system is (5, 2). Check this
solution in both equations (1) and (2).
Solution set: {(5, 2)}
5-6
Solving a System by Substitution
To solve the system graphically, solve both equations
for y:
Graph both Y1 and Y2 in the
standard window to find
that their point of
intersection is (5, 2).
5-7
Solving a System by Elimination
Solve the system.
Multiply both sides of equation (1) by 2, and then
multiply both sides of equation (2) by 3.
Add equations (3) and (4), then solve for x.
5-8
Solving a System by Elimination
Substitute 4 for x equation (1), then solve for y.
The solution of the system is (4, –3). Check this
solution in both equations (1) and (2).
5-9
Solving a System by Elimination
Solution set: {(4, –3)}
5-10
Solving a System by Elimination
The graph confirms that the solution set is {(4, –3)}.
5-11
Solving an Inconsistent System
Solve the system.
Multiply both sides of equation (1) by 2, then add the
resulting equation to equation (2).
14 x  6 y  10
(3)
14 x  6 y  10
(2)
0  20 False
The system is inconsistent.
Solution set: ø
5-12
Solving an Inconsistent System
The graphs of the equations are parallel and never
intersect.
5-13
Solving a System with Infinitely Many Solutions
Solve the system.
Multiply both sides of equation (2) by 3, then add the
resulting equation to equation (1).
The result indicates that the equations of the original
system are equivalent. Any ordered pair that satisfies
either equation will satisfy the system.
5-14
Solving a System with Infinitely Many Solutions
From equation (2), we have
The solution set (with x arbitrary) is {(x, 3x – 8)}.
From equation (2), we have
The solution set (with y arbitrary) is
5-15
Solving a System with Infinitely Many Solutions
The graphs of the two equations coincide.
5-16
Assignment
• Sec. 5.1: pp. 504-505 (7-27 odd, 31-39 odd)
– Write the system and solve. Use solution sets.
– Use the indicated method to solve.
– When a system has an infinite number of solutions,
use y as the arbitrary variable.
• Solve in terms of y.
– Due tomorrow, Wednesday, 22 April 2015.
5-17
Assignment: Sec. 5.1: pp. 504-505 (7-27 odd, 31-39 odd)
Due tomorrow, Wednesday, 22 April 2015.
For 7-17, solve by substitution.
4 x  3y  13
7)
9)
x  y  5
x  5y  8
11) 8 x  10y  22
7 x  y  10
13)
3y  x  10
3x  y  6
15)
2 x  6y  18
17)
29  5 y  3 x
For 19-27, solve by elimination.
19) 3 x  y  4
21)
x  3y  12
23)
x  6y
5 x  7y  6
10 x  3y  46
27)
3y  5 x  6
xy 2
2 x  3y  7
5 x  4 y  17
25) 6 x  7y  2  0
x y
 4
7 x  6y  26  0
2 3
3 x 3y

 15
2
2
5-18
Assignment: Sec. 5.1: pp. 504-505 (7-27 odd, 31-39 odd)
Due tomorrow, Wednesday, 22 April 2015.
Solve each system. State whether it is inconsistent or has infinitely many
solutions. If the system has infinitely many solutions, write the solution set
with y arbitrary.
31) 9 x  5 y  1
 18 x  10 y  1
35) 5 x  5 y  3  0
x  y  12  0
33) 4 x  y  9
 8 x  2y  18
37) 7 x  2y  6
14 x  4y  12
39) Get from the book.
5-19