Use a Graphing Calculator to Solve a Systems of Equations

Solving a systems of equations can become messy after a while. This is especially true when you have three or more equations to be solved. Try using your graphing calculator instead by following these easy steps.

Steps

Graphically (Maximum of 2 Variables, Linear and Nonlinear)

  1. Convert the equations into y= form In order to graph an equation on your calculator, the left side of the equation must only contain y. This is pretty easy to do, especially if you are dealing with linear equations. All you have to do is solve your equation for y. If it is a linear systems in standard form (Ax+By=C), then you can just write down the following equation, subbing in for the constants A, B, and C: y=(c-ax)/b or you can use the second method. If your equations are in slope intercept form, you already have the equation in y= form.
  2. Plug the equations into your calculator. Press the y= button, usually the top left button on your calculator, and type in your first equation under y1 and your second equation under y2. Since you can only use 2 variables with this method, you shouldn't need more than 2 equations. If instead of y=, you have something else, make sure you are in function mode by pressing the mode button and making sure "Func" is highlighted.
  3. Make sure all plots are turned off and other equations cleared. On the TI83, plots that are turned on are highlighted in the y= menu. If there are any on, simply move your cursor over them and press enter. You should also scroll down on the y= screen to make sure there are no other equations set to graph. If there are you have two options:
    • If you might need the equation later, move your cursor to the "=" sign and press enter to unselect it. Only the entries with their "=" signs highlighted will graph, but your equation will still be entered if you need it. To make the equation graph again, move your cursor onto the "=" sign again and press enter to highlight it.
    • If you won't need the equation later, move your cursor to the entry (not the "=" sign) and press clear. You can't retrieve the equation if you do this, so be sure you want to clear it before you do this.
  4. Choose an appropriate window. Guess at what your answer might be. You want to be your answer is on the screen. Press the "window" button, usually the button just to the right of the y= button. The xmin should be considerably less than your guess and the xmax should be considerably higher. Then, instead of worrying about your y=, just press the "zoom" button just to the right of the "window" button then scroll down to "ZoomFit" and press enter. In a few seconds your graph should appear on the screen. If you can't see the intersection of the graphs, adjust the window until you can.
  5. Find the intersection of the graphs. Press "2nd" then "Trace" to bring you to the "Calculate" menu. Scroll down to intersect and press enter. It should prompt "First curve?" Make sure your cursor is on one of the graphs and press enter. It should then prompt "Second curve?" Make sure your cursor is on the other graph and press enter. It will prompt "Guess?" The easiest way to do this for equations that only intersect once is to press enter again. If they intersect twice on the same page, either enter a value that is close to the x value where they intersect (You could use your estimated value for x in step 4) or move your cursor near where they intersect. Remember: it doesn't have to be perfect; your calculator is just trying to make sure it finds the right intersection point for you. It doesn't really need an accurate guess. To navigate on the graph screen:
    • Press the up and down buttons to switch between graphs. This will not change your x value.
    • Press the left and right buttons to move along the graph. Your x value change will be incremental, your y change will be the output of the x value, so your cursor may move faster some times and slower others.
  6. Write down your solution. On the bottom of your screen, your calculator should say "x=a number" and "y=a number." These are your answers. Copy them down to the desired accuracy.

Using Matrices (No Limit on Variables, Only Linear)

  1. Convert the equations into standard form. In order for this method to work, your equations must be linear in the form Ax+By+Cz...=F. For equations in slope-intercept form (y=mx+b), just plug in the constants into this formula: y-mx=b. For others, get all of your variables onto the left side and your constants on the right and simplify. Remember which order your variables were in for when you interpret the final matrix.
  2. Go to the matrix editor. On the TI83, the keystrokes are "2nd" then "x-1" to enter the matrix screen. On other calculators, look for the word "matrix." Move your cursor right until you reach the edit screen. Scroll down to a matrix you aren't using and press enter.
  3. Enter your data. Look at the top. Your cursor should be over a number, then there should be an "x" sign followed by another number. The first number corresponds to the number of rows in the matrix and the second number corresponds to the number of columns. Type in the number of variables for the rows and type in the number of variables plus 1 for the columns. Press enter after typing in each number. Once you've entered the row and column numbers, you will be taken to the data part. At the bottom, the "coordinates" of the cell you are entering will be shown. The first number corresponds to the row number and the second to the column number. Make sure that anything with a 1 in the rows column should be for the first equation, anything with a 2 is for the second, etc. and that anything with a 1 should be for the first variable, anything with a 2 is for the second variable etc. Be sure all coefficients for a variable are in the same column. To enter the data, just type in the first coefficient (a coefficient is the number before the variable) of the first equation, then the second coefficient, then the third coefficient etc. The last number in the row should be the constant on the right side of the equation. After typing in each number press enter. At the end of a row, press enter to be taken to the next row. Type in the equations in their corresponding rows in this manner.
  4. Find the solution. Quit out of the editor. On the TI83 press "2nd" then "delete." Use the quit button instead of pressing clear or you might end up deleting the data cell your cursor was over. From the home screen, go back to the matrix screen (not the editor), and move your cursor to the right until you reach the "Math" menu. Scroll down to "rref" (be sure you don't go to "ref" by mistake) and press enter. Go to the matrix screen again, but this time just scroll down to the matrix you put your data into and press enter. Your home screen should say "rref([your matrix name here]." Press enter to get the resultant matrix. Your answer is in the column farthest to the right. The top number corresponds to the first variable, the next number down corresponds to the second variable, etc.

Tips

  • Different models of calculators have different keystrokes, but for the most part, keystrokes on the TI83 editions work on the TI84 editions and the TI83 keystrokes work on all versions of the calculator including the plus and silver editions.

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