Transcript Systems of Equations PowerPointx

Let’s Play a Game…
Pull out a piece of paper (rip it in half)
Write down a Number (any number you want)
Call it x ( x = ____ )
Now Pick Another Number
Call it y ( y = ____ )
What does x + y = ?
What does x – y = ?
Example:
x=3
y=5
x+y=8
x – y = -2
Example:
SUBSTITUTION!
(You can use either equation)
x+y=7
x – y = 13
x+y=7
2x + 0 = 20 This method
(10) + y = 7
is called -10
2x = 20
-10
x = 10 ELIMINATION
y = -3
Once you know x,
how do you find y?
Can I solve real situations using equations?
ᶱ
ᶱ
ᶱ
YES!
Define your variables.
Create equations based on the
information provided.
You can solve for both variables using:
ᶱ Elimination
ᶱ Substitution
ᶱ Graphs
Step 1: Put the equations in
Standard Form.
Step 2: Determine which
variable to eliminate.
Standard Form: Ax + By = C
Look for variables that have the
same coefficient. You may need to
multiply to ensure they are the same.
Step 3: Add or subtract the
equations.
Solve for the variable.
Step 4: Plug back in to find
the other variable.
Substitute the value of the variable
into the equation.
Step 5: Check your
solution.
Substitute your ordered pair into
BOTH equations.
You can do
this on your
calculator
Mrs. Drawbaugh decided to purchase
candy for her whole class as a treat. She
bought Smarties and Dum-Dum lollipops
as “brain food” for their next exam.
Each bag of Smarties cost $7.00
(including tax). The bag of Dum-Dum
lollipops cost $8.50 (including tax). She
ended up spending $60.50 on her
purchase of 8 items.
Smarties: $7.00
Dum-Dum lollipops: $8.50
Number of Smarties Number of Dum-Dum Total Cost for 8 Items
Bags
lollipops Bags
($)
0
1
2
3
4
5
6
7
8
8
7
6
5
4
3
2
1
0
$68.00
$66.50
$65.00
$63.50
$62.00
$60.50
$59.00
$57.50
$56.00
Determine the number of Smartie Bags and
Dum-Dum lollipop Bags Mrs. Drawbaugh
bought with $60.50.
Smarties: $7.00
Dum-Dum lollipops: $8.50
8 bags of candy bought total
𝑥 = # 𝑜𝑓 𝑠𝑚𝑎𝑟𝑡𝑖𝑒𝑠 𝑏𝑎𝑔𝑠
𝑦 = # 𝑜𝑓 𝐷𝑢𝑚 − 𝐷𝑢𝑚 𝑙𝑜𝑙𝑙𝑖𝑝𝑜𝑝 𝑏𝑎𝑔𝑠
𝑥+𝑦 =8
$7.00𝑥 + $8.50𝑦 = $60.50
Elimination
𝑥 + 𝑦 = 8 (Multiply by -7.00 to eliminate the x-values)
$7.00𝑥 + $8.50𝑦 = $60.50 (Keep the equation as-is)
−7.00𝑥 − 7.00𝑦 = −56
7.00𝑥 + 8.50𝑦 = 60.50
1.5𝑦 = 4.5
𝑦=3
𝑥, 𝑦 = (5,3)
𝑥 = # 𝑜𝑓 𝑠𝑚𝑎𝑟𝑡𝑖𝑒𝑠 𝑏𝑎𝑔𝑠
𝑦 = # 𝑜𝑓 𝐷𝑢𝑚 − 𝐷𝑢𝑚 𝑙𝑜𝑙𝑙𝑖𝑝𝑜𝑝 𝑏𝑎𝑔𝑠
𝑥 + (3) = 8
𝑥=5
5 Smarties bags
3 Dum-Dum lollipop bags
Substitution
𝑥+𝑦 =8
𝑦 = −𝑥 + 8
$7.00𝑥 + $8.50𝑦 = $60.50
7.00𝑥 + 8.50(−𝑥 + 8) = 60.50
7.00𝑥 − 8.50𝑥 + 68 = 60.50
−1.50𝑥 = −7.50
𝑥=5
Substitute 5 for x in
either equation to
solve for y
𝑦 = −𝑥 + 8
𝑦 = −(5) + 8
𝑦=3
𝑥, 𝑦 = (5,3)
𝑥 = # 𝑜𝑓 𝑠𝑚𝑎𝑟𝑡𝑖𝑒𝑠 𝑏𝑎𝑔𝑠
𝑦 = # 𝑜𝑓 𝐷𝑢𝑚 − 𝐷𝑢𝑚 𝑙𝑜𝑙𝑙𝑖𝑝𝑜𝑝 𝑏𝑎𝑔𝑠
5 Smarties bags
3 Dum-Dum lollipop bags
Graphing
𝑥+𝑦=8
$7.00𝑥 + $8.50𝑦 = $60.50
𝑥 = # 𝑜𝑓 𝑠𝑚𝑎𝑟𝑡𝑖𝑒𝑠 𝑏𝑎𝑔𝑠
𝑦 = # 𝑜𝑓 𝐷𝑢𝑚 − 𝐷𝑢𝑚 𝑙𝑜𝑙𝑙𝑖𝑝𝑜𝑝 𝑏𝑎𝑔𝑠
5 Smarties bags
3 Dum-Dum lollipop bags
You can do
this on your
calculator
Business
Expense
Models
Package Comparisons
Significant Events
2010
Economics
MAXIMUM WEIGHT
=
BREAKING POINT
When solving a System of LINEAR equations you will have either:
One Solution
No Solution
(intersects at one point)
(Never intersects)
Infinite Solutions
(Intersects infinitely– Same Line)