Solving Two Simultaneous Linear Equations

To be solved, word or story problems must be translated into equations with algebraic expressions that contain constants and variables. Simple word problems may not need variables at all. Some are more conveniently soved with the introduction of one or more variables. Usually, the number of resulting equations equals the number of the introduced variables. The simplest of the multivariable/equation problems is that with two variables and two linear equations. As always, there are many ways to approach the same problem. Below we look into two such ways of solving a linear system of two simultaneous equations.

The first approach is based on the fact that, if two equations are satisfied, so is their difference (or the sum, for that matter.) Also, this is one of Euclid's common notions that if two equal quantities are multiplied by the same factor, the results remain equal. Using that, we cam multiply the two equations by such factors as to make the coefficients by the same variable equal. Subtracting then one equation from the other eliminates that variable.

(Note: In the applet below, all underlied words and numbers can be clicked on. Click, click, click ... and see what happens.)

// foreground color // background color // information pieces start with '@', followed by // a single letter from {v,a,f,e}, followed by ": ". // v - variable - anything that may change on a click // variables must precede all other pieces // f - formulation - word problem with embedded variables. // Many formulations are possible for a single problem. // Only one is shown at a time // e - equation - equation is like a formulation with a // potential provision to, e.g., enclose negative numbers // into parentheses // a - answer - like an equation, but allowed to have a parsable // portion. Parsables are written in the reverse Polish notations // with variables and operations separated by a comma // d - directive // define color, skip line // // whatever it is, a variable can be of the three types: // i = integer - on click changes up (right of center) and down (left of center) // a = attribute - any clickable and changeable word // v = variable - like an attribute but without related attributes // all 'name's below are 1 letter from [A-Ba-b0-9] // I, name, initial value, min, max // equations

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Another approach is to express one of the unknowns in terms of the other from one of the equations and substitute that into the other equation. The result is one equation with one unknown. We solve this first and then find the first unknown. See for yourself:

// foreground color // background color // information pieces start with '@', followed by // a single letter from {v,a,f,e}, followed by ": ". // v - variable - anything that may change on a click // variables must precede all other pieces // f - formulation - word problem with embedded variables. // Many formulations are possible for a single problem. // Only one is shown at a time // e - equation - equation is like a formulation with a // potential provision to, e.g., enclose negative numbers // into parentheses // a - answer - like an equation, but allowed to have a parsable // portion. Parsables are written in the reverse Polish notations // with variables and operations separated by a comma // d - directive // define color, skip line // // whatever it is, a variable can be of the three types: // i = integer - on click changes up (right of center) and down (left of center) // a = attribute - any clickable and changeable word // v = variable - like an attribute but without related attributes // all 'name's below are 1 letter from [A-Ba-b0-9] // I, name, initial value, min, max // equations

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

(There are many more word problems discussed and solved at this site. The math tutorial continues with a similar approach over several additional examples.)


Word Problems

  1. Problems of class a + x = b
  2. Problems of class a · x = b
  3. From Word Problem to Equation
  4. Problems of class x / a = b / c
  5. Problems of class x = a + b
  6. Problems of class x = a + b (II)
  7. Problems of class x = a × b
  8. Solving Two Simultaneous Linear Equations

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