Elimination Using Multiplication
Multiplying Both Equations to Simplify the System
For some systems of equations, one equation must be multiplied
by a fraction in order to make elimination by addition or
subtraction possible. Since multiplication of integers is easier,
both equations are multiplied by nonzero numbers so that the
coefficients of a variable in the equations become equal (or
opposite).
Example 1
Solve the system of equations.
2x + 9y = 7
3x + 7y = 4
Solution
One approach is to multiply the first equation by , and then subtract the
resulting equation from the second one. This method works, but
involves fractional arithmetic. Another approach is to multiply
the first equation by 3 and the second by 2, to get an equivalent
system of equations.
3(2x + 9y = 7 ) 

6x + 27y = 21 

2(3x + 7y = 4 ) 

6x + 14y = 8 



6x + 27y = 21 



(  ) 6x + 14y = 8 
Subtract the equations. 


0 + 13y = 13 



13y = 13 



y = 1 
Divide each side by 13. 
Now substitute 1 for y in the second equation.
6x + 14y = 8 

6x + 14(1) = 8 
Replace y with 1. 
6x = 6 

x = 1 

The solution is ( 1, 1).
