How to solve three linear equations in three variables by elimination

Three equations can look like a lot to keep track of at once. Let’s eliminate the same variable from two pairs of equations so we can work with a smaller system, then use that result to finish.

Eliminate the same variable twice

(1)-x-5y+z=17(2)-5x-5y+5z=5(3)2x+5y-3z=-10

The y coefficients in equations (1) and (3) are opposites, so adding them cancels y. Adding equations (2) and (3) cancels y again and gives a second equation in x and z.

(1)+(3)⟶x-2z=7(2)+(3)⟶-3x+2z=-5

Solve the smaller system and substitute back

Add the two new equations to cancel z. Then put the value of x into the first reduced equation and use x and z in original equation (1) to find y.

-2x=2,x=-1-1-2z=7,z=-41-5y-4=17,y=-4

The solution is (-1,-4,-4). Substituting the three values into the original left sides gives 17, 5, and -10, which match all three right sides.

What happens when the equations are inconsistent?

(1)3a-3b+4c=-23(2)a+2b-3c=25(3)4a-b+c=25

The coefficient of a is 1 in equation (2). Multiply that equation by 3 and subtract it from (1), then multiply equation (2) by 4 and subtract it from (3).

(1)-3(2)⟶-9b+13c=-98(3)-4(2)⟶-9b+13c=-75

The same expression would have to equal both -98 and -75. Subtracting the equations gives 0=-23, so the original system has no solution.

Parallel lines from the two reduced equationsThe two reduced equations draw parallel lines in the b c plane. One has constant negative ninety-eight and the other negative seventy-five, so they have no common point.bc−9b + 13c = −98−9b + 13c = −75
The two reduced equations have the same slope and different intercepts.

For more practice with the smaller system, see how to solve a system of two linear equations.

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