Solve to get x, y and z

Question

Solve: 2x+y+z=3

            2x-y-z=9

            X+y-z=0

Answer 1

Hi, Victoria. When you are dealing with a system of equations in three variables, you want to see first if there is anyway that you can eliminate two variables in one fell swoop. Remember that for any one given equation, you can multiply both sides by the same number without making the equation untrue. Also, remember that you can add or subtract two equations.

 

If we add the first and second equations, we will eliminate both the x and the y.

 

(2x + y + z) + (2x - y - z) = (3) + (9)

2x + y + z + 2x - y - z = 3 + 9

4x = 12

x = 3

 

So, x must equal 3.

 

Now, if we put 3 into each equation, we have the following equations:

6 + y + z = 3

6 - y - z = 9

3 + y - z = 0

 

Now, we can move all the numbers to the right side:

y + z = -3

-y - z = 3

y - z = -3

 

Now, we can try to eliminate another variable. If we add the second two equations, we can eliminate y.

 

(-y - z) + (y - z) = (3) + (-3)

-y -z + y - z = 0

-2z = 0

z = 0 (because, if we divide both sides by -2 to isolate z, 0/[-2] = 0)

 

So, now we have x = 3 & z = 0.

 

Now, pick any of the original equations and plug these values of x & z in, and you will get y.

 

2x + y + z = 3

2(3) + y + 0 = 3

6 + y + 0 = 3

6 + y = 3

y = -3

 

So, now we have x = 3, y = -3, & z = 0.

These are the only x, y, and z values that work for all 3 equations.

Answer 2

2x + y + z = 3

2x - y - z = 9

x + y - z = 0

 

4x = 12             add top and middle

x = 3

 

3x + 2y = 3       add top and bottom

9 + 2y = 3

 

Can you solve for y and z?

Answer 3

Label the equations (1), (2) and (3) for convenience

2x+y+z=3   (1)
2x-y-z=9     (2)
X+y-z=0     (3)

 

Add (1) + (2) as this will eliminate both y and z from the resulting equation.

Always check the equations for the simplest eliminations possible. Sometimes you can eliminate two variables as in this case.

 

4x=12 -->  x=3

 

Now add equations (2) and (3) which will eliminate y

3x-2z=9

Substitute for x and solve for z

3(3)-2z=9  -->z=0

 

Now substitute for x and z in one of the equations to find y. I'll use (1)

2(3)+y+0=3 -->y=-3

Solution is (3,-3,0)