Chapter 9
Unit Question
How do we solve linear systems?
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Warm-up to follow…
Solve using Addition
x  y  10
x  y  22
2 x  3 y  12
 2 x  y  20
 5 x  4 y  15
2 x  4 y  6
2x  32
2
2
x  16
4 y  32
4
4
y 8
 5 x  4 y  15
 2 x  4 y  6
(16)  y  10
 16
 16
y  6
(16,6)
 2 x  (8)  20
8 8
 2 x  12
2 2
x  6
(6,8)
 7 x  21
7 7
x  3
2(3)  4 y  6
y0
(3,0)
Solve using Addition
6x  5 y  2
12 x  8 y  5
Problems? Let’s talk about it...
Section 5
Essential Question
How do we solve linear systems
using multiplication?
Homework Clicky
Just like last time…with times!
1)
1a)
Line up the equations
Multiply, if needed
EXAMPLE
x+y=5
3x + 4y = 12
x+y=5
x + -3 = 5
+3 +3
x=8
2)
“Add” straight down
3)
Finish solving for the
remaining variable
y = -3
4)
Substitute answer into
an original equation
So…the solution is the
ordered pair…(8, -3)
-3x + -3y = -15
3x + 4y = 12
Find the 2 unknowns!
Step 1 …
line ‘em up
2x + 3y = 4
-x + 4y = 9
2x + 3y = 4
2x + 3(2) = 4
Step 1a
multiply by 2
2x + 3y = 4
-2x + 8y = 18
2x + 6 = 4
-6 -6
Step 2 “add”
straight down
11y = 22
11 11
Step 3 solve
for ‘y’
Step 4
substitute and
solve for’x’
y=2
2x = -2
2 2
x = -1
(-1, 2)
Find the 2 unknowns!
4x + 5y = 7
6x – 2y = -18
Multiply the 1st by 2
AND
Multiply the 2nd by 5…
8x + 10y = 14
30x – 10y = -90
38x = -76
38
38
x = -2
4x + 5y = 7
4(-2) + 5y = 7
-8 +5y = 7
+8
+8
5y = 15
5
5
y=3
(-2, 3)
Find the 2 unknowns a different way!
4x + 5y = 7
6x – 2y = -18
Multiply the 1st by 3
AND
Multiply the 2nd by -2…
12x + 15y = 21
-12x + 4y = 36
19y = 57
19
19
y=3
4x + 5y = 7
4x + 5(3) = 7
4x +15 = 7
-15 -15
4x = -8
4
4
x = -2
(-2, 3)
EITHER WAY SAME ANSWER
The 8th grade team spent $51.50 on 4 staplers and 5 boxes of paper…
The 7th grade team spent $65.50 on 10 staplers and 1 box of paper…
What’s the cost of 1 stapler and 1 box of paper?
Let x = cost of stapler
y = cost of paper
4x + 5y = 51.50
-50x – 5y = -327.50
-46x = -276.00
-46
-46
x = $6.00
4x + 5y = 51.50
10x + y = 65.50
Multiply 2nd equation by -5
1 Stapler + 1 Paper = $11.50
4x + 5y = 51.50
4(6) + 5y = 51.50
24 + 5y = 51.50
-24
-24
5y = 27.50
5
5
y = $5.50
Homework
• HoffmaSheet 9 – 5