Systems of Equations
Systems of Equations
• Finding all values of all variables that make the
equations true.
What values make an equation true?
7x-3y=12
What values make an equation true?
y=4x+1
What values make all equations true?
y=4x+1
7x-3y=12
What values make all equations true?
y=4x+1
7x-3y=12
(x,y)=(-3,-11)
e) (1,1)
“y=4x+1” is pronounced “the
number y IS THE SAME
NUMBER AS 4x+1”
Solving with substitution
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7x-3y=12
y=4x+1
7x-3(4x+1)=12
7x-12x-3=12
-5x-3=12
+3=+3
-5x=15
x=-3
Original Problem
Original Problem
Anywhere I see a y, I can write (4x+1)
Distributive property
(un)distributing
add the same thing to both sides
Divide both sides by -5
But wait, I’m not done!
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7x-3y=12
y=4x+1
x=-3
y=4(-3)+1
y=-11
(x,y)=(-3,-11)
Original Problem
Original Problem
Solution so far.
Anywhere I see an “x”, I can write “(-3)”
“2x+3y=3” is pronounced “2x+3y IS THE
SAME NUMBER AS 3”
b
Infinitely many solutions
• If you wind up with something that is ALWAYS true, you
have infinitely many solutions
• 2x+y=1
GIVEN
• 4x+2y=2
GIVEN
• (-2)(2x+y)=(-2)(1)Multiply both sides by -2
• -4x-2y=-2
New Equation
• 4x+2y=2
Add to original
• -------------• 0x+0y=0
• 0=0
Always true  infinitely many solutions
No solutions
• If you wind up with something that is NEVER true,
you have infinitely many solutions
• 2x+y=3
GIVEN
• 4x+2y=2
GIVEN
• (-2)(2x+y)=(-2)(3) Multiply both sides by -2
• -4x-2y=-6
New Equation
• 4x+2y=2
Add to original
• -------------• 0x+0y=-4
• 0=-4
Never true  No solutions
Find the solution: 2x-3y = 1
6y-4x = -2
A)
B)
C)
D)
x = 2, y = 1
x =0, y = -1/3
x = 6, y = 5/3
Both A & B and infinitely many
other solutions
E) Both B & C and infinitely many
other solutions
Solution
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2x-3y = 1
6y-4x = -2
(2)(2x-3y) =(2)(1)
multiply both sides
4x-6y =2
6y-4x = -2
add to original
-------------------4x-4x+6y-6y=2-2
0=0, infinitely many solutions
But is the answer D (A&B) or E (B&C)?
Plug in A (2,1) and check: 2(2)-3(1)=1 A is a solution
The answer is D