What are the Values of the Variables?

Question

What are the vaules of the variables x, y, and z.

-2x+y=-5

-2y+z=-4

2x-2z=10

Answer 1

x is 3

y is 1

z is -2

Answer 2

One way to solve systems of equations is by adding equations together.

In this case, the first and the third will cancel out the x term, so let's try that.

-2x + 1y + 0z = -5

2x + 0y - 2z = 10

0x + 1y - 2z = 5

y = 5 + 2z

Plugging this into the second equation gives:

-2y + z = -4

-2(5 + 2z) + z = -4

-10 - 4z + z = -4

-3z = 6

z = -2

Now that we have a value for z, plug that back into the equation for y:

y = 5 + 2z

y = 5 - 4

y = 1

Plugging the y or z values back into their original x equation should yield the same answer:

-2x + y = -5

-2x + 1 = -5

-2x = -6

x = 3

2x - 2z = 10

2x + 4 = 10

2x = 6

x = 3

Good!  Now that we have values, make sure to plug them back into the original 3 equations to make sure everything works.  To be honest, the first time I tried answering this I had a mistake and only 2 of the 3 worked out!

-2x + y = -5

-6 + 1 = -5  Yes!

-2y + z = -4

-2 + -2 = -4  Yes!

2x - 2z = 10

6 - -4 = 10  Yes!

ALWAYS make sure to check your work like this if you can.