System of Equations

Question

x + y + z = 4

x - 2y - z = 1

2z - y - 2z = -1

 

 

Answer 1

x + y + z = 4                  eq1
x - 2y - z = 1                   eq2
2x - y - 2z = -1                eq3

 

 

We have here a system of equations. We need to solve for three different variables.  Each equation will have the same value of x, y, and z.  When you plug them in, each equation is satisfied.

 

We can use the substitution and eliminations methods to solve.

 

Substitute eq1 into eq2 and eq3 so that we can reduce the number of equations and variables. From eq1,

 

z = 4 - x - y

 

 

x - 2y - (4 - x - y) = 1

x - 2y - 4 + x + y = 1

2x - y = 5             new2

 

2x - y - 2(4 - x - y) = -1

2x - y - 8 + 2x + 2y = -1

4x + y = 7              new eq3

 

 

Now we have two equations to work with.

 

2x - y = 5              new eq2

4x + y = 7             new eq3

 

 

We can add the equations to eliminate the y terms.

 

6x = 12

 

 x = 2

 

 

Substitute this value of x into new eq2 to solve for y.

 

2(2) - y = 5

 

4 - y = 5

 

   -y = 1

 

    y = -1

 

 

Substitute the values of x and y into the original eq1 to solve for z.

 

z = 4 - x - y

 

z = 4 - 2 - (-1)

 

z = 4 - 2 + 1

 

z = 3

 

 

Your solutions are

 

x = 2

y = -1

z = 3