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).
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, 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.
