2. History of the
simultaneous 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.
3. 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.
4. First thing you have to know to solve
simultaneous equations
• -,- = +
• +,+ = +
• -,+ = -
• +,- = -
• Example: 2-(-3)=2+3=5
• 3+(-2)=3-2=1
6. Now...
This cannot be
4x-3y=4 solved either.
This also has two unknown numbers.
7. 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.
8. Simultaneous equation
Can get one single answer if we put the two previous
equations together.
A single value for x and a single value for y.
Those values will be the only ones that work in both
equations at the same time.
9. 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.
10. Two kind of ways of solving simultaneous equation
Elimination
Substitution
11. Elimination
First way to solve simultaneous equation
Adding or subtracting one side that will leave
0 at the side you add or subtract and only one
unknown at the other side.
12. 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.
13. 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
14. 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
15. Substitution
Another way to solve a simultaneous
equation.
To transform one equation into
x=something or y =something and
substituting that something into the
other equation’s x or y.
16. Example of using substitution
Substituting 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
17. Example of using substitution
Substituting 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
18. Real life application
Air traffic control tower:(To prevent 2
planes from crashing into each other)
Economics:(To identify the relation between
2 goods)
Restaurant:(To choose the best menu for a
meal)
