Solving Simultaneous Equations
Substitution Method
There are many times in physics where you obtain 2 or more simultaneous equations. With 2
equations, you can solve for the unknowns by hand with reasonable care and you will probably
not make a mistake. For example, you might solve the following equations with the substitution
method.
3x – 2y = 7
-4x + y = -6
The second equation can be solved for y, and then substituted into the first equation to obtain a
single equation with a single unknown.
y = -6 + 4x
3x – 2(-6 + 4x) = 7
3x + 12 – 8x = 7
-5x + 12 = 7
-5x = -5
x=1
This value for x can be put back into either of the equations and solved for y.
3(1) – 2y = 7
3 – 2y = 7
-2y = 4
y = -2
Now you can throw both numbers back into the original two equations to check your work.
3(1) – 2(-2) = 7 (yes!)
-4(1) + -2 = -6 (yes!)
Gaussian Elimination
Another way to solve the original pair of equations is with Gaussian elimination. Start by picking
a variable to eliminate. Let’s pick x to eliminate. Multiply the first equation by the coefficient of
x in the second equation and multiply the second equation by the opposite of the coefficient of x
in the first equation.
-4[3x – 2y = 7]
-3[-4x + y = -6]
-12x + 8y = -28
12x + -3y = 18
Now add the two equations, then solve for y.
5y = -10
y = -2
This process can be repeated to eliminate y and solve for x, or you can simply place y into one of
the equations and solve for x.
Matrices
With 3 or more equations, the number of steps for either of the above methods grows
significantly. This means that there is a significant likelihood of making a mistake and wasting
lots of time on boring math when you could be doing more physics, playing Frisbee, or listening
to Vanilla Ice. Once you demonstrate that you can solve a system of equations by hand, there is
no reason not to have a calculator or computer do it for you. The way to do this is to first learn
how to put a system of equations in augmented matrix form. The original two equations could be
written in augmented matrix form as follows:
3  2  7 
  4 1   6


If you can figure out how to put this matrix into your calculator and tell it how to solve for the
unknowns, then you will improve your life. This will of course depend on the make and model of
your calculator. The only tricks are to put the variables in a consistent order on the left side of the
equation, shove constants to the right side of the equation, and correctly input implicit 1s and 0s.
For example, let’s get the following system of equations ready for input into a matrix:
x + 3y = 8 + 5y
-3z + 4x = 5y
z + 7y -9 = 0
The variables should be put in order on the left side and the constant shoved to the right side:
x – 2y = 8
4x – 5y – 3z = 0
7y + z = 9
Now let’s put in the implicit 1s and 0s.
1x + -2y + 0z = 8
4x + -5y + -3z = 0
0x + 7y + 1z = 9
Now this can be put into augmented matrix form:
1  2 0 
4  5  3 

0 7
1 
8
0
9
This can be put into a calculator and solved for the unknowns. If you are a math geek, you might
call it “row-reduced echelon form.”
TI Matrix Help

TI-83 Plus & TI 84 (on the TI-83, MATRIX is on the keyboard, and is not in second
position)
1. Create Matrix
a. Press [2nd] [MATRX].
b. “Arrow” over to EDIT.
c. “Arrow” down to the letter of your choice (you are giving your matrix a name
like [A] or [B] or [C] or…) and press [ENTER].
d. Select size of matrix by first entering the number of rows and then entering the
number of columns; i.e., for a 3 4 matrix press [3] [ENTER] [4] [ENTER].
Note that if the matrix you are entering is an augmented matrix, the last column
of your matrix will represent the constants after the “equals” sign of your system
of equations.
e. Insert the entries of the matrix, pressing the [ENTER] key after each entry.
f. When finished entering all entries, press [2nd] [QUIT].
2. Putting the Matrix into Row-Reduced Echelon Form
Press [2nd] [MATRX].
“Arrow” over to MATH.
“Arrow” down to B:rref( and press [ENTER].
You will see rref( on your screen now. Enter the matrix name; i.e., press
[2nd] [MATRX], “arrow” down to the name of your matrix, and then press
[ENTER].
e. Close your parentheses and press [ENTER]. You now have your matrix in RowReduced Echelon Form.
a.
b.
c.
d.

TI-86
1. Create Matrix
a. Press [2nd] [MATRX].
b. Press [F2] to EDIT a matrix.
c. Enter a name for your matrix. If you are using a name that hasn’t been used for a
matrix before, then enter it in using letters A through Z (in light blue) on the
calculator. Press the [ENTER] key when you are ready to continue.
d. Select size of matrix by first entering the number of rows and then entering the
number of columns; i.e., for a 3 4 matrix press [3] [ENTER] [4] [ENTER].
Note that if the matrix you are entering is an augmented matrix, the last column
of your matrix will represent the constants after the “equals” sign of your system
of equations.
e. Insert the entries of the matrix, pressing the [ENTER] key after each entry.
f. When finished entering all entries, press [EXIT].
2. Putting the Matrix into Row-Reduced Echelon Form
Press [2nd] [MATRX].
Press [F4] to use the OPS menu.
Press [F5] to select rref .
Press [EXIT] and then [F1] to get the NAMES of your matrices. Select your
matrix name from the on-screen menu by pressing the appropriate function key
just below the screen.
e. Press [ENTER]. You now have your matrix in Row-Reduced Echelon Form.
a.
b.
c.
d.

TI-89/92
1. Create Matrix
a. Press [APPS].
b. “Arrow” down to Data/Matrix Editor, press [ENTER].
c. “Arrow” down to “New” and press [ENTER]. In the “New” window, make sure
the “Type” is “Matrix,” make sure to name your “Variable,” and adjust the “Row
dimension” and “Column dimension” as desired. Press [ENTER] when ready to
continue.
d. Insert the entries of the matrix, pressing the [ENTER] key after each entry. Note
that if the matrix you are entering is an augmented matrix, the last column of
your matrix will represent the constants after the “equals” sign of your system of
equations.
e. When finished entering all entries, press [HOME].
2. Putting the Matrix into Row-Reduced Echelon Form
Press [2nd] [MATH].
“Arrow” down to Matrix and press [ENTER].
“Arrow” down to rref( and press [ENTER].
Enter the name of your matrix. Close the parentheses when finished entering the
name of the matrix.
e. Press [ENTER]. You now have your matrix in Row-Reduced Echelon Form.
a.
b.
c.
d.
Thanks to Lorna TenEyck for her help with this document.