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 1

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

### Step 2

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."

### Step 3

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."

### Step 4

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 1

Press "On."

### Step 2

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.

### Step 3

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 1

Press "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.

### Step 2

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).

### Step 3

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

## Solving The Equations

### Step 1

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

### Step 2

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."

### Step 3

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.