Simultaneous Equations:
Equations that must be solved at the same time are simultaneous equations.
E.g.
2x + 3y = -9
x + 4y = 6
We use three different methods to solve simultaneous equations. They are:
- elimination method
- substitution method
- graphical method
Elimination Method
In this method, we must get rid of one variable in order to find the other.
E.g.1
x + y = 6 ---
1
x - y = 2 ---
2
If we add the two equations, we can remove y.
1 +
2 => 2x = 8
x = 4
Sub in
1=> 4 + y = 6
y = 2
Solutions are x = 4 and y = 2.
E.g.2
2x + y = 6 ---
1
3x - 2y = 2 ------
1
To remove y, multiply the first equation by 2 and then add the two equations together.
1 X 2 => 4x + 2y = 12 ------
3
2 +
1 => 7x = 14
x = 2
Sub in
1
4 + y = 6
y = 2
The solutions are x = 2 and y = 2.
E.g.3
2x + 3y = 1 ---
1
3x - 2y = 8 ---
2
In this case, to eliminate y, the first equation must be multiplied by 2 and the second equation must be
multiplied by 3.
1 X 2 => 4x + 6y = 2 ---
3
2 X 3 => 9x - 6y = 24 ---
4
3 +
4 => 13x = 26
x =2
Sub in
1=> 4 + 3y = 1
-4 => 3y = -3
y = -1
The solutions are x = 2 and y = -1.
Substitution Method
We get y in terms of x or vice versa from one equation, and put that in the other.
E.g.1
x + y = 6 ---
1
x - y = 2 ---
2
From
1 => x = (6 - y)
Sub this in
2 => 6 - y - y = 2
6 - 2y = 2
-6 => -2y = -4
:--2 => y = 2
Sub in
1 => x + 2 = 6
x = 4
Solutions are x = 4 and y = 2.
E.g.2
2x + y = 6 ---
1
3x - 2y = 2 ------
2
From
1 => y = (6 - 2x)
Sub this in
2 => 3x - 2(6 - 2x) = 2
3x - 12 + 4x = 2
7x - 12 = 2
+ 12 => 7x = 14
x = 2
Sub in
1 => 4 + y = 6
-4 => y = 2
The solutions are x = 2 and y = 2.
E.g.3
2x + 3y = 1 ---
1
3x - 2y = 8 ---
2
From
1 - 3y => 2x = (1 - 3y)
x = (1 - 3y)/2
Sub in
2 => 3(1 - 3y)/2 - 2y = 8
X 2 => 3(1 - 3y) - 4y = 16
3 - 9y - 4y = 16
3 - 13y = 16
-3 => -13y = 13
y = -1
Sub in
1 => 2x - 3 = 1
+ 3 => 2x = 4
x = 2
The solutions are x = 2 and y = -1.
Graphical Method
In this method, two straight lines are drawn for each equation. Then the point where the two lines intersect at is noted. The coordinates of this point are the solutions of the equations.
E.g.1
2x + y = 8 ---
1
y -x = 1 ---
2
1 => y = 8 - 2x
2 => y = x + 1
Rearrange the two equations in the form of y = mx + c and draw two lines for them on the same grid.
The coordinates of the point of intersection are x = 3 and y = 2.
So, the solutions are x = 3 and y = 2.
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