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Algebra - Two Variables

Variable Substitution

You can solve a system of equations with a method called substitution. Here’s how to use it.

You have two equations, so “solve” one equation by writing A in terms of B. That is, use the rules of algebra to rearrange the equation to get “A =” on one side. Then substitute this expression into the other equation wherever variable A appears. This results in one equation and one unknown.


y = 4x

x + y = 90

What are the values of x and y?


The first equation y = 4x tells you that “4x” is another name for y.  
Substitute 4x for y: x + y = 90
  x + 4x = 90
  5x = 90
Divide both sides by 5: 5x/5 = 90/5
  x = 18
To find y, substitute 18 for x in either of the original equations: y = 4x
y = 4(18)
  y = 72
Substitute the values of x and y into the other equation: x + y = 90 ?
  18 + 72 = 90 ? Yes!



Solve the following system of two equations:

Q + N = 33.

25Q + 5N = 505. 

Look at the equations.

What constant should we use?

-5(Q + N) = -5(33)
Let’s multiply the second equation by -5: –5Q –5N = -165
Add both equations together: 25Q + 5N  =  505 
  -5Q – 5N = -165
  20Q =  340
Solve for Q: 20Q/20 = 340/20
  Q = 17
Substitute Q = 17 to solve for N: 17 + N = 33
  N = 33 – 17 = 16
Check the result with both: 25(17) + 5(16) = 505? Yes!
  17 + 16 = 33? Yes!
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