# A quick way to solve a system of 3 equations?

• jumbogala
In summary, the person seeking help has a system of three equations and three variables. Their calculus professor was able to solve it quickly by setting some variables to zero and using algebraic operations to eliminate terms and solve for the remaining variables. The person seeking help is unsure of the exact steps their professor took, but one possibility is combining equations and using substitution to solve for each variable. Another possibility is using algebraic operations to eliminate terms and solve for each variable.
jumbogala

## Homework Statement

I need to quickly be able to solve something like this:

A + B + C = -3
6A + 3B + C = -4
8A - 4B - 2C = 22

My calculus prof set some things to zero and was able to solve it really fast that way. He didn't use matrices. I can't figure out now what he did... can anyone help?

## The Attempt at a Solution

I think he was making each of the variables 0 at one point, but I tried that and it definitely didn't work, haha.

Your description of what your prof did wasn't too clear, but he might have done this:
Add -6 times the first equation to the second equation, then add -8 times the first equation to the third equation. These two operations eliminate the A terms from the second and third equation.
The system now looks like this:
A + B + C = -3
- 3B - 5C = 14
- 12B - 10C = 46

Now add -4 times the second equation to the third, which eliminates B from the third equation, which allows you to solve for C, which you can use in the 2nd equation to find B, and which you can use in the 1st equation to find A.

If that wasn't it, here's another possibility -- Add the 1st and 2nd equations to the 3rd, which gives you 15A = 15, or A = 1. Then substitute that value for A in the 1st and 2nd equation and subtract the 2nd from the first, from which you can solve for B, and eventually C.

## 1. How do I solve a system of 3 equations quickly?

There are several methods for solving a system of 3 equations quickly, but one of the most efficient ways is by using the Gauss-Jordan elimination method. This involves converting the system of equations into an augmented matrix and using row operations to simplify and solve for the variables.

## 2. Can I use a calculator to solve a system of 3 equations quickly?

Yes, many scientific and graphing calculators have built-in features for solving systems of equations. You can input the equations and use the calculator's function to solve for the variables.

## 3. Is there a specific order in which I should solve the equations?

The order in which you solve the equations does not affect the final solution. However, it is often helpful to start with the simplest equation or the one that has the fewest variables, as it may make the process faster and easier.

## 4. How do I know if a system of 3 equations has a solution?

A system of 3 equations will have a solution if the number of equations is equal to the number of variables and the equations are not parallel or coincident. This can also be determined by graphing the equations and checking for intersections.

## 5. Can I use substitution or elimination to solve a system of 3 equations quickly?

Yes, both substitution and elimination are effective methods for solving systems of equations, but they may take longer than the Gauss-Jordan elimination method. It is recommended to use these methods if the equations are already in a simplified form.

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