From Algebra I to the SAT Math and beyond, the ability to solve simultaneous first-order (i.e. no exponents) equations can be a lifesaver. While solving for two variables (x,y) is doable, solving for three variables (x,y,z) can be very time consuming. Plus, more variables means a greater risk of careless mistakes. Luckily, the TI-89 calculator can make quick work of three-variable simultaneous equations, and this article will take you through the process step-by-step.

## Getting The Coefficients

Step 1Determine all of the variables used in the equations (x, y, z)

Rearrange each equation so that it's in the form "ax + by + cz = d" where a, b, c & d are constants. Also, if one of the variables doesn't appear in one of the equations, set its coefficient to "0", i.e. "6y + 4x = 15" would be written "4x + 6y + 0z = 15."

On a piece of scratch paper, list the equations in a column, one on top of the other. Make sure that the variables in each equation are in the same order, i.e. first "x," then "y," then "z."

Erase or scratch out all of the variables, leaving only the coefficients and the constants in the last column.

## Entering The Coefficient Matrix Into The TI-89

Step 1Press "On."

Press "APPS" button. Press "6" for the "Data/Matrix Editor." Press "3" for "New." Press "->" for "Type:". Press "2" for "Matrix." Under "Variable," type "Coefficient." Press the "Enter" button. Under "Row Dimension," press "3." Under "Col Dimension," press "3." Press the "Enter" button twice.

Fill in the empty 3 x 3 matrix with the coefficients from the equations. Once finished, press "2nd" button, then the "Exit" button.

## Entering The Constant Matrix Into The TI-89

Step 1Press "APPS" button. Press "6" button for the "Data/Matrix Editor." Press "3" for "New." Press "->" for "Type:". Press "2" for "Matrix." Under "Variable," type "Constant." Press the "Enter" button. Under "Row Dimension," press "3." Under "Col Dimension," press "1." Press the "Enter" button twice.

Fill in the empty 3 x 1 matrix with the column of constants from the equations (the number on the right side of the "=" sign in each equation).

Once finished, press "2nd" and then the "Exit" button.

## Solving The Equations

Step 1Press "2nd." Press "5" and then "4" and then "5." The word "SIMULT(" should appear on the lower right side of the screen.

Press "2nd."Press the "-" button ("Var-Links" will be written in yellow above the button). Scroll down to "Coefficient" and press "Enter." Press "," button. Press "2nd." Press "-" button for Var-Link. Scroll down to "Constant" and press "Enter." Press ")" and then "Enter."

Translate the resulting 3 x 1 matrix in the final solution: the top number equals x, the middle number equals y and the bottom number equals z.