Solve the following systems in three variables using linear combinations.

Question

11x+15y+5z=1 

3x+4y+z=-2

7x+13y+3z=3

Answer 1

From the second equation, z=-2-3x-4y.

Substitute for z in the other two equations:

11x+15y+5(-2-3x-4y)=1, 

7x+13y+3(-2-3x-4y)=3.

Simplifying these two equations:

11x+15y-10-15x-20y=1⇒-4x-5y=11,

7x+13y-6-9x-12y=3⇒-2x+y=9.

Multiply the equation above by -2 on each side:

4x-2y=-18 and add to the other equation:

-4x-5y=11

-7y=-7, y=1.

Substituting for y in -2x+y=9:

-2x+1=9, -2x=8, x=-4.

Finally, z=-2-3x-4y=-2+12-4=6.

So the solution is x=-4, y=1, z=6.