Solving Systems of Linear Equations. The SHARP EL-546W Calculator

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1 Solving Systems of Linear Equations with The SHARP EL-546W Calculator The Sharp EL-546W calculator is capable of solving: systems of linear equations with two unknowns ( 2 2 systems), and systems of linear equations with three unknowns ( 3 3 systems). Standard Form for Systems of Equations To use the calculator to solve a system of equations, the system must be written in standard form so that the coefficients can be easily identified. The common textbook forms for systems of two and three equations are as follows: For a 2 2 system, the standard form is: a 11 x 1 a 12 x 2 b 1 a 21 x 1 a 22 x 2 b 2 For a 3 3 system, the standard form is: a 11 x 1 a 12 x 2 a 13 x 3 b 1 a 21 x 1 a 22 x 2 a 23 x 3 b 2 a 31 x 1 a 32 x 2 a 33 x 3 b 3 The SHARP Conventions The conventions used for entering systems of equations into the SHARP calculator are based on the following standard forms: Two Simultaneous Linear Equations with two unknowns For a 2 2 system, the standard Sharp form is Example a 1 x b 1 y c 1 8x 5y 45 a 2 x b 2 y c 2 4x 2y 10 Three Simultaneous Linear Equations with three unknowns For a 3 3 system, the standard Sharp form is a 1 x b 1 y c 1 z d 1 a 2 x b 2 y c 2 z d 2 a 3 x b 3 y c 3 z d 3 Example 7x 25y 8z 19 2x 15y z 210 x 15y 3z 10 Sharp EL546W - Systems of Equations (SharpEL546WSystems.tex)...1

2 Solutions to Linear Systems with the Sharp EL546W To solve a system of two or three simultaneous linear equations: Set the Calculator Mode Enter the coefficients and constants Setting the Mode For 2 equations (2-VLE):...Enter: MODE 2 0 Display prompts for the first coefficient: a1? For 3 equations (3-VLE):...Enter MODE 2 1 Display prompts for the first coefficient: a1? Examples Use your calculator to solve each of the following systems of linear equations. 1. Solve the two variable system:... Compare to Standard form:... 8x 5y 45 4x 2y 10. a 1 x b 1 y c 1 a 2 x b 2 y c 2 Set the Mode to 2-VLE...Enter: MODE 2 0 Enter the coefficients in response to the prompts: a1? 8 ent b1? 5 ent c1? 45 ent a2? 4 ent b2? 2 ent c2? 10/ ent Display is: x 35 ent Display is: y 65 ent Display is: det 4 Answer:... The solution values for 8x 5y 45 4x 2y 10 are x 35 and y 65 det 4 is the value of the determinant of coefficients Sharp EL546W - Systems of Equations (SharpEL546WSystems.tex)...2

3 2. Solve the three variable system:... 7x 25y 8z 19 2x 15y z 210 x 15y 3z 10. Compare to Standard form:... a 1 x b 1 y c 1 z d 1 a 2 x b 2 y c 2 z d 2 a 3 x b 3 y c 3 z d 3 Set the Mode to 3-VLE...Enter: MODE 2 1 Enter the coefficients in response to the prompts: a1? 7 ent b1? 25 ent c1? 8/ ent d1? 19 ent etc. d3? 10 ent Display is: x ent Display is: y ent Display is: z ent Display is: det 475 Answer:... The solution values to and z x 25y 8z 19 2x 15y z 210 x 15y 3z 10 are x ; y det 475 is the value of the determinant of coefficients Sharp EL546W - Systems of Equations (SharpEL546WSystems.tex)...3

4 Exercise Now solve each of the following using the Sharp EL546W calculator. 1. 3x 5y 90 x 2y 10 Solution is : x , y x 15y 90 3y 26 (note the coefficient of 0 for the x term in the 2nd equation) Solution is : x , y x 25y 19 2x 15y 100 Solution is : x , y (remember to rearrange in standard form) 4. 8a 3b 18c 19 a b c 210 a 15c 10 (note the coefficient of b 3 0 for the b term in the 3rd equation) Be careful not to confuse the variables a, b and c with the coefficients a 1 8,, b 1 3,etc. The EL546W always identifies the variables as X, Y and Z. Solution is : a , b , c x 25y 11 7x y The calculator provides an Error 2 message. Do you know why? There is no solution for this system of equations since the lines are parallel lines and do not intersect. Sharp EL546W - Systems of Equations (SharpEL546WSystems.tex)...4

5 6. 4x 5y 11 8x 22 10y The calculator provides an Error 2 message. Do you know why? There are infinite solutions for this system of equations since both equations describe the same line x 25y 8z 2x 15y z 210 (remember to rearrange into standard form). x 15y 100. Solution is : x , y , z a 4b 200 a 2c 15. 3b 2c a 10. Solution is : a , b , c b c a 50. a b c 0 2a c Solution is : a 25.0, b 15.0, c a b c d 50 a b c 2d 0 Your calculator is not capable of solving a 4 4 2a c 4d 20 system of equations, but Scientific Notebook is! b c 3d 40 Solution is : a 50, b 30, c 80, d 50, Remember, to reset the calculator to NORMAL MODE MODE 0 Sharp EL546W - Systems of Equations (SharpEL546WSystems.tex)...5

6 Solving Systems of Equations with Scientific Notebook The Sharp EL-546W calculator is capable of solving only linear systems of two and three equations. Other mathematical tools have additional capabilities. Scientific Notebook can solve systems of equations: with any number of equations and unknowns. Example: a b c d 50. a b c 2d 0 2a c 4d 20 b c 3d 40 with linear and/or non-linear systems. Example: 2x 3y 5. x 2 y 2 16 with equations having non-algebraic terms Example: y e x 2x 2 3y Example: y sin2x y 2x 2 If you are viewing this as an online Scientific Notebook document, select the following link for details on Solving Systems of Equations with Scientific Notebook. Copy the URL: Solving Systems of Equations with Scientific Notebook (file size: 92k) or and paste it into the dialogue box under FileOpen Location in Scientific Notebook. Sharp EL546W - Systems of Equations (SharpEL546WSystems.tex)...6

7 Notes: The solution to a system of equations is the set of values for the variables that will satisfy ALL of the equations in the system simultaneously. It is possible for a linear system of equations to have: exactly 1 unique solution no solutions at all (called an inconsistent system) an infinite solution set (called a dependent system) Online Documents The online Scientific Notebook version of this document can be accessed from within Scientific Notebook by pasting the following URL into the dialogue under FileOpen Location.: The online Adobe PDF version of this document can be accessed from any web browser, with the URL: Sharp EL546W - Systems of Equations (SharpEL546WSystems.tex)...7

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