Transcript Chapter 4 – Systems of Linear Equations

8-1: Systems of Linear Equations
(2 Variables)
y
y
x
x
y
y=½x+3
2y = x + 6
Infinite solutions (x,y)
y=½x-3
y=½x+2
x
No Solutions
y = ½ x –3
y = (-2/3)x + 2
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: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
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?
Three Equations & Three Unknowns
Step 1: Eliminate 1 variable (the same variable) from two sets of equations
Step 2: Solve the resulting 2 equations/2 unknowns w/substitution or addition
Step 3: Solve for the 3rd unknown using found variables and the 3rd equation.
Find solution: {5, -3, -2}
x + y + z = 0
x + 2y – 3z = 5
3x + 4y + 2z = -1
Find solution: {1, 1, 2}
x +
z = 3
x + 2y – z = 1
2x – y + z = 3
Application: Investments
Kelly has $20,000 to invest As her financial planner,
you recommend that she diversify into three investments:
• Treasury bills that yield 5% simple interest
• Treasure bonds that yield 7% simple interest, and
• Corporate bonds that yield 10% simple interest.
Kelly wishes to earn $1390 per year in income. Also, Kelly wants her investment
In Treasury bills to be $3000 more than her investment in corporate bonds.
How much money should Kelly place in each investment?
8.6 Non-Linear Systems of Equations
For non-linear equations, there is no general methodology.
Only experience using the techniques in your ‘mathematical toolbox’
can help you.
Graphing the equations, this can give you quick insight into howmany
intersection points / solutions the system has.
3x – y = -2
2x2 – y = 0
Using substitution,
Y = 2x2
3x - 2x2 = -2
2x2 -3x -2 = 0
( 2x + 1)(x – 2) = 0
X = -1/2 or x = 2
3(-1/2) – y = -2
y = 1/2
or
3(2) –y = -2
y=8
8.7 Graphing Systems of Linear Inequalities
Graph: x + y < 4
y
x y
0 4
1 3
Graph the system of inequalities
2x – y < 2
x y
x
y
x + 2y > 6
0 -2
0 3
1 0
2 2
y
x
x
Step 1: Graph the line
Step 2: Decide: Dashed or Solid
Step 3: Choose a test point on 1 side
Step 4: Plug in test point & check
Step 5: Shade the TRUE side.
Test Point (0,0)  0 + 0 < 4 --- TRUE
Test and Shade
See in-class example
8.7 Graphing Systems of Non-Linear Inequalities