History of thesimultaneous equations• The most ancient records of the use of simultaneous equations were found in Samaria as far back as 2000 BC.• No one know who exactly invented simultaneous equations.• The Babylonians used simultaneous equations a lot as far back as 1800 BC.
Main Vocabulary• Simultaneous: Both at the same time• Elimination will tell you about these two words later.• Substitution• Absolute Value: The numerical value of a real number without regard to its sign.
First thing you have to know to solve simultaneous equations • -,- = + • +,+ = + • -,+ = - • +,- = - • Example: 2-(-3)=2+3=5 • 3+(-2)=3-2=1
Before learning about simultaneous equation 2x+3y=20Unknown numbers
Now... This cannot be 4x-3y=4 solved either.This also has two unknown numbers.
Third thing you have to know before learning about simultaneous equation What you do to the one side, do it to the other side. That will make the previous equation and the new equation the same.
Simultaneous equationCan get one single answer if we put the two previousequations together.A single value for x and a single value for y.Those values will be the only ones that work in bothequations at the same time.
What is a simultaneous equation?• A set of equations that have more than one value.• Can solve both equations at the same time.• More than two equations with 2 values(x and y)• x and y are unknown and has to be found.
Two kind of ways of solving simultaneous equation Elimination Substitution
EliminationFirst way to solve simultaneous equationAdding or subtracting one side that will leave0 at the side you add or subtract and only oneunknown at the other side.
Elimination Can change the equation by multiplying the same number at both sides. Even if the number is changed, the value will remain the same as long as you multiply the same number for both sides. Once you get one of the variables, then you can get the other variable.
Example of using elimination Eliminating Variable X x2 x2 x2 2x+3y=20 4x+6y=40 4x-3y=4 4x-3y=4 4x+6y=40 4x+6(4)=4x+24=40- 4x-3y=4 4x=40-24, 4x=16, x=4 0 +9y=36 9y=36, y=4
Example of using elimination Eliminating Variable Y 2x+3y=20 2x+3y=20+ 4x-3y=4 2(4)+3y=20 6x =24, x=4 8+3y=20 3y=20-8 3y=12, y=4
SubstitutionAnother way to solve a simultaneousequation.To transform one equation intox=something or y =something andsubstituting that something into theother equation’s x or y.
Example of using substitutionSubstituting x into y 2x+3y=20, 2x=20-3y, x=10-1.5y 4x-3y=4, 4(10-1.5y)-3y=4, 40-6y-3y=4, -9y=4-40, -9y=-36, 9y=36, y=4 2x+3(4)=20 2x+12=20 2x=20-12 2x=8, x=4
Example of using substitutionSubstituting y into x 2x+3y=20, 3y=20-2x, y=20 - 2x 3 3 20 2x 4x-20+2x=4, 4x-3y=4, 4x-3( - )=4, 4x 3 3 +2x=4+20, 6x=24, x=4 2(4)+3y=20 8+3y=20 3y=20-8 3y=12, y=4
Real life applicationAir traﬃc control tower:(To prevent 2planes from crashing into each other)Economics:(To identify the relation between2 goods)Restaurant:(To choose the best menu for ameal)