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### Course: Algebra (all content)>Unit 5

Lesson 9: Systems with three variables

# Solving linear systems with 3 variables

Sal solves a system of three variables using elimination. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• How do you solve for 4 variables?
• If you have 4 equations and 4 variables, then you can use the method used in this video. This method works as long as you have at least 1 equation for each variable.
• do you always have to check your answer or do you just do it so we can see the answer is right?
• It is definitely better to check your answers when doing problems like these because having to do so much work can sometimes make us miss something as little as forgetting a negative sign. Also, make sure to check it for each equation because sometimes it equals one or two of them but not all three.
• at why is he adding all the numbers when he could be subtracting?
• so that he can eliminate the z variable.
• At , he says to eliminate the y variable by multiplying by 7. The equations are:
16x+7y=-2
8x-y=-10
Wouldn't it be easier to eliminate the x variable by multiplying the second equation by -2?
• Absolutely! You would be on your way to get the correct value for y if you chose your method. There are many approaches to eliminating the variable terms when solving using the elimination system.
• How do you solve 2 equations with 3 unknowns?
• You cannot unless you have some other information. As a general rule you need 2 equations to solve for 2 unknown, 3 equations to solve for 3, etc.

If you have a specific question you are referring to post it and we'll see if we can help.
• I added all three equations to eliminate the y variable but ended up getting a wrong answer. Is adding all the equations a valid operation?
• There is actually a way to solve this with just a graphing calculator!
Here are the steps.
1. Turn on your graphing calculator. (It needs to be a TI-83 or better)
2. click 2nd, matrix.
3. click to the right until you are on the setting, EDIT.
4. select 1 of the matrices. It will bring up the matrix size on the top row and the matrix at the bottom.
5. change the matrix size to 3 x 4.
6. click to the right until you are in the matrix itself.
7. The equation we are doing is x+y-3z= -10, x-y+2z= 3, 2x+y-z= -6. Take the coefficients and plug them into the matrix. This includes the values in the last column.
8. click 2nd QUIT.
9. now click 2nd matrix again but now press right to go to the setting MATH.
scroll down the list until you find the option RREF. (not REF).
10. click RREF then go back to 2nd matrix then press the letter of the 3 x 4 matrix.
11. press enter!
Your calculator will form a row of answers in the shape of the identity matrix with all of the values for the variables in the last column.
Enjoy!
if the matrix does not become the identity matrix, it means that the planes are either parallel, overlapping, intersecting a line of points or never all touching at once.
If my instruction were confusing, there are actually videos on this that can explain it better than I can.
I hope I helped! :)
After learning this, I never had to do these again!
• Is it possible to scale all 3 equations at once to save time?
(1 vote)
• Yes. But unless you're scaling them mentally as you copy them down (which I often do), it doesn't really save much time. Another problem is that you may end up working with much larger numbers than necessary, because you need to find the least common multiple of 3 numbers rather than 2. Considering you will probably need to multiply at least one of the equations by a negative as well, the chance of making a careless mistake is pretty high.
• Could you expand on scaling up when needing to eliminate a variable. Where can I find more examples of this?
• To "scale up," one would simply multiply both sides of the equation by a value. The reason why one would scale up is to rewrite the equation so that a variable term will eliminate when it is added to another equation. A simpler example is to consider the system, 2x + 3y = 2 and x + 4y = 5. You can "scale up" the second equation by -2, that is in other words, multiply both sides of the second equation by -2 so that the "x" term becomes "-2x." Since the second equation has a "-2x" and the first equation has a "2x" the x-terms will eliminate when the equations are combined.

Good luck!