Systems of Equations by Elimination
To solve a linear system of equations by the
method of elimination:
1) write both of the equations in standard form, x
term and y term on the left-hand side of the equation and constant term on
the right-hand side of the equation,
2) place the equations in column form (one equation under the other) and
pre-add (in your head) the x-terms and the y-terms from both equation to
see if one or both of the like terms will be eliminated
if neither the x terms, nor the y terms will be eliminated, then you will
have to multiply one or both of the equations by some integer so that the
coefficients of either the x terms or the y terms will be additive
once you have made appropriate changes to one or more of the
lines by multiplying then actually add the two equations, eliminating
either the x terms or the y terms,
3) step 2 will result in an equation in one variable, solve this equation
for the variable,
4) use the value found in step 3 to find the value of the remaining
variable, you may use either equation to find the second value,
5) check the solution in both equations to be sure that the solution
works in both equations.
Solve the system of equations
3x 2y = 19 and x +
y = 8
(Notice that this is the same system that I solved previously by
substitution. I want to show you the differences in the two methods.
Actually, only the first two steps are different. Steps 3-5 are the same
for both methods.)
3x 2y = 19
x + y = 8
pre-adding I would
get 4x y = 27, so I must multiply one of the lines by some integer so
that the coefficients of one of the variable terms are additive inverses.
I choose to eliminate the y terms since multiplying the second line by 2
would make 2y and + 2y additive inverses.
(2)x +(2)y = (2)8
2y = 16
now place the equations once again in column form and add
3x 2y = 19
2x + 2y = 16
5x = 35
x = 7
now use this value and
substitute it into either of the original equation to find the value of y.
3(7) 2y = 19
21 2y = 19
2y = 19 21
2y = 2
y = 1
the solution set is the point (
7, 1 )
Check: the solution set should work in both of the
equations 3x 2y = 19 and x + y = 8.
Since 3(7) 2(1) = 19 and (7) + (1) = 8, then the point ( 7, 1) is the
solution to this system of equations.
General Algebra Tips
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