Simultaneous equations consist of two or more equations that are true at the same time. Consider the following example:
Claire and Laura are sisters; we know that
(i)  Claire is the elder sister, 

(ii)  their ages added together give 20 years, 
(iii)  the difference between their ages is 2 years. 
Let x = Claire's age, in years and y = Laura's age, in years.
x + y = 20 
x  y = 2 
This is an example of a pair of simultaneous equations.
In this section we consider two methods of solving pairs of simultaneous equations like these.
Use a graph to solve the simultaneous equations:
x + y  =  20 
x  y  =  2 
x + y  =  20 
y  =  20  x 
x  y  =  2  
x  =  y + 2  
x  2  =  y  
or  
y  =  x  2 
Use a graph to solve the simultaneous equations:
x + 2y  =  18 
3x  y  =  5 
x + 2y  =  18  
2y  =  18  x  
y  =  
y  = 
 
y  = 

3x  y  =  5  
3x  =  y + 5  
3x  5  =  y  
or  
y  =  3x  5 
Now draw these two graphs:
The lines cross at the point with coordinates (4, 7), so the solution is x = 4, y = 7.Solve the simultaneous equations:
x + 2y  =  29  (1) 
x + y  =  18  (2) 
Note that the equations have been numbered (1) and (2).
Start with equation (2)
x + y  =  18 
y  =  18  x 
Now replace y in equation (1) using y = 18  x
x + 2y  =  29 
x + 2(18  x)  =  29 
x + 36  2x  =  29 
36  x  =  29 
36  =  29 + x 
36  29  =  x 
x  =  7 
Finally, using y = 18  x gives
y  =  18  7 
y  =  11 
So the solution is x = 7, y = 11
Take equation (2) away from equation (1).
x + 2y  =  29  (1) 
x + y  =  18  (2) 
y  =  11  (1)  (2) 
In equation (2), replace y with 11.
x + 11  =  18 
x  =  18  11 
x  =  7 
So the solution is x = 7, y = 11
Solve the simultaneous equations:
2x + 3y  =  28  (1) 
x + y  =  11  (2) 
From equation (2)
x + y  =  11 
y  =  11  x 
Substitute this into equation (1)
2x + 3(11  x)  =  28 
2x + 33  3x  =  28 
33  x  =  28 
33  =  28 + x 
33  28  =  x 
x  =  5 
Finally, use y = 11  x
y  =  11  x 
y  =  11  5 
y  =  6 
So the solution is x = 5, y = 6
Subtract 2 × equation (2) from equation (1).
2x + 3y  =  28  (1) 
2x + 2y  =  22  2 × (2) 
y  =  6  (1)  2 × (2) 
Now replace y in equation (2) with 6.
x + 6  =  11 
x  =  11  6 
x  =  5 
So the solution is x = 5, y = 6
Solve the simultaneous equations:
x  2y  =  8  (1) 
2x + y  =  21  (2) 
From equation (2)
2x + y  =  21 
y  =  21  2x 
Substitute this into equation (1)
x  2y  =  8 
x  2(21  2x)  =  8 
x  42 + 4x  =  8 
5x  42  =  8 
5x  =  8 + 42 
5x  =  50 
x  =  10 
Now substitute this into y = 21  2x
y  =  21  2 × 10 
y  =  21  20 
y  =  1 
So the solution is x = 10, y = 1
Subtract 2 × equation (1) from equation (2).
2x + y  =  21  (2) 
2x  4y  =  16  2 × (1) 
5y  =  5  (2)  2 × (1) 
y  =  1 
Now replace this in equation (1).
x  2y  =  8 
x  =  8 + 2 
x  =  10 
So the solution is x = 10, y = 1