Archive for the ‘Lesson 13: Simultaneous Linear Equations’ Category


Lesson 13: Simultaneous Linear Equations. A Joke (III)

March 10, 2011

Algebra is a language, you have to know its words, rules, how to translate,….

What do you think a maths teacher would do in an English class?

Here you are a true history

About application of mathematics in linguistics

A teacher of English was ill and a teacher of mathematics replaced him. He began to compose a table of irregular verbs:

Then he said:
– Okay, I mark this form as  x . Then it’s possible to compose the proportion:


Lesson 13: Simultaneous Linear Equations: Solving Word Problems (II)

March 5, 2011

3. Solving word problems with simultaneous linear equations

Systems of linear equations are a powerful tool to solve problems. Many problems lead themselves to being solved with systems of linear equations. In “real life”, these problems can be incredibly complex. This is one reason why linear algebra (the study of linear systems and related concepts) is a very important branch of mathematics. However, you will face with much simpler problems in Secondary Education. Here you are some typical examples.

Steps for Solving Word Problems

 1. Read the problem. Relax. Underline important words and values.

 2. (Read again). Introduce variables:

 Let x = describe in words

Let y = describe in words

3.(Read again). Make a Table:


  Description of x Description of y Total
Description for combined quantities ax     bya’x     b’y cc’
4. Write down the system of equations (from table):
ax + by  = c a’x + b’y = c’

 5.Solve for  x and  y using substitution, equalizing or elimination.

6.Write the answer in words including proper units.

7.Check the answers not only in the system but also in the context of the problem.

Here you are a good site to practise.

First you have got solved examples. At the bottom of the page you will find a generator of problems. But you can jump right to the exercises if you click on the link that says Jump right to the exercises! at the beginning.

  1. Click on “new problem” to get started.
  2. On these exercises, you will not key in your answer.
  3. However, you can check to see if your answer is correct on answer. The solution is explained in detail.

Finally here you are a video where is explained how to solve a numeric word problem with simultaneous linear equations.


Lesson 13: Simultaneous Linear Equations (I)

February 28, 2011


Vocabulary on simultaneous linear equations

1. Linear equations

 A linear equation in two unknowns is an equation of first degree with two unknowns.

 Any linear equation can be expressed in its standard form

ax + by = c

where a, b and c are known numbers.

 A solution of a linear equation in two unknowns is a pair of values that verify the equation (they make it true)

A linear equation in two unknowns has infinite solutions.

We call system (or pair) of linear equations or simultaneous equations in two unknowns to a collection of such equations.

ax + by = c

a’x + b’y = c’

 A solution of the system is any pair of numbers that satisfies them simultaneously.

 Solving a pair of simultaneous equations is to look for the values of the unknowns which make all the equations true simultaneously

 2. Methods of solving simultaneous equations

 We now focus on various methods of solving simultaneous equations. All these methods have the same philosophy: to obtain from the two equations another equation only in one unknown. Once we solve it is easy to determine the value of the other unknown.

 2.1. Substitution method

 Here it is the method of substitution:

  • Solve one of the equations for one unknown in terms of the other.
  • Then, substitute that in the other equation.
  • That will yield one equation in one unknown, which we can solve.
  • We obtain the other unknown substituting in any equation of the system

In this page you can introduce any system of linear equations, solve it by yourself and find a complete explanation about how to do it by using substitution method. Unfortunately, this algebra solver don’t use elimination method.

Here you are a video on substitution method.

2.2. Equalizing Method

 This method is not explained in Anglo-Saxon countries, probably because it is a slight variation of substitution method.

 Here it is the method of “Equalizing”:

  • Solve both equations for the same unknown in terms of the other.
  • Then, equalize the expressions in terms of the other.
  • That will yield one equation in one unknown, which we can solve.
  • We obtain the other unknown substituting in any equation of the system

 Sorry, I couldn’t find a web page where the equalizing method were explained and that had interactive exercises.

 2.3. Elimination method

 This method is also named addition method.

 As the name suggest, this method tries to eliminate variables until there’s only 1 variable left by adding the equations.

 Here it is a general strategy for solving simultaneous equations:

  • Look at the equations and try to find 2 equations which has the same coefficient (plus or minus) for the same variables.
  • When the coefficients don’t have the same coefficients (plus or minus), multiply the equations so that the coefficients of that unknown are the same
  • Add or subtract  the equations vertically, and that unknown will cancel(we eliminate the unknown)
  • We will then have one equation in one unknown, which we can solve.
  • To solve for y, the other unknown substitute the value of x in one of the original equations.

 This page is highly recommendable because you can find several videos on solving simultaneous equations by using elimination method, but they have got a higher level. The video below is elemental but good to improve your listening

In the page I am going to recommend you, you have interactive exercises on solving simultaneous equations by using elimination method. It has a difficulty because they always subtract instead of add up as we do.

Take into account that if you subtract an equation from other you have to change every sign in the LHS and in the RHS to subtract correcly.

Click on

Go down the page to find the exercise. Get an example and try it by yourself. If you have got any difficulty, click on the buttons below. Finally, use only example 1. Example 2 is harder to understand for you.