Transcript Chapter 4 – Systems of Linear Equations

Ch4: Systems of Linear Equations
y
y
x
x
y
y=½x+3
2y = x + 6
Consistent
Dependent Equations
Many solutions (x,y)
y=½x-3
y=½x+2
Inconsistent
No Solutions
x
y = ½ x –3
y = (-2/3)x + 2
Consistent
Independent Equations
One Solution (x, y)
Solving Systems: Graphical Method
Step 1: Graph both equations
Step 2: Find the point of intersection
x + 2y = 7
x–y= 4
y
x
0
1
x
y
3½
3
x
0
1
y
-4
-3
Solution: (5, 1)
Solving Systems: Substitution Method
Step 1: Solve for x or y in 1 equation
Step 2: Substitute into the 2nd equation
Step 3: Solve algebraically to find 1 variable
Step 4: Plug the ‘found variable’ back in and
solve for the second variable.
x + 2y = 7
x -y = 4
y
Step 1: x – y = 4
x=y+4
x
[add y to both sides]
Step 2: x + 2y = 7
(y + 4) + 2y = 7
Step 3:
3y + 4
3y = 3
y=1
=7
Step 4:
x=y+4
x = (1) + 4
x=5
Solution
(5, 1)
Solving Systems:Elimination/Addition
Method
x + y = 10
x–y=8
2x
x
9
5 2A + 3B = -1
-3 3A + 5B = -2
= 18
2
x
2
x + 2y = 7
x - y =4
2
=
9
+ y = 10
+ y = 10
y =1
Solution (9, 1)
x + 2y = 7
2x - 2y = 8
3x
= 15
3
3
x
=
5
x–y=4
5-y =4
y=1
Solution (5, 1)
10A + 15B = -5
-9A - 15B = 6
A = 1
2A + 3B = -1
2(1) + 3B = -1
2 + 3B = -1
3B = -3
B = -1
Practice
1.
2.
Solving using the elimination/addition method
2x - y = 4
5x + y = 3
Solving using the substitution method
x + 2y = 2
3x + 4y = 0
3. Solve using any way you choose
2x + 5y = 2
-3x + 4y = 20
Answers
1.
(1, -2)
2.
(-4, 3)
3.
(-4, 2)
Applications
Write a system of equations for each scenario and solve:
1 Bank Account Problem: Carlos has 2 bank accounts. He has seven times
as much in his savings account as in his checking account. In all, he has
$3,200 in the bank. Find out how much Carlos has in each account.
2. Coins in a Jar: There are 93 coins in a jar. The coins are quarters and dimes
All together the coins total $18.60. How many quarters and dimes are in the jar?
3. Dimensions of a Rectangle: A soccer field has a perimeter of 320 yards. The
length measures 40 yards more than its width. What are the field dimensions?
4. Acid Mixture (Revisited): How many ounces of a 5% hydrochloric acid and 20%
hydrochloric acid must be combined to get 10 oz of solution that is 12.5% acid?
Section 4.3
Systems of Linear Equations and
Problem Solving
Suppose you mix an amount of 25% acid solution
with an amount of 60% acid solution. You then
calculate the acid strength of the resulting acid
mixture. For which of the following results should
you suspect an error in your calculations?
a.) 14%
b.) 32%
c.) 55%
© 2008 Pearson Prentice Hall. All rights reserved
Event Ticket Problem
Hilton University Drama Club sold 311 tickets for a play.
Student tickets cost 50 cents each, nonstudent tickets cost $1.50.
If total receipts were $385.50,
find how many tickets of eqach type were sold.
Finding Break-Even
A manufacturing company recently purchased $3000 worth of new
equipment to offer new personalized stationery to its customers.
The cost of producing a package of personalized stationery is $3.00 and it is sold for
$5.50. Find the number of packages that must be sold for the company to break even
Let x = number of packages of personalized stationery
C(x) = total cost for producing x packages of stationery
R(x) = total revenue for selling x packages of stationery