Modelling and control by Anthony Rossiter

SIMULTANEOUS EQUATIONS AND STRAIGHT LINES

 

These videos give an introduction to straight lines. Where do they come from and, for example, what does the formulation y=mx+c represent?

Having understood what a straight line represents, the videos move onto concepts of simultaneous equations, that is, when two straight lines meet. The videos begin by giving simple everyday examples of scenarios where straight lines occur and where the intersection is meaningful.

Finally, the videos introduce simple techniques for solving for the intersection points, again focussing on commonsense approaches rather than abstract concepts. Use the left hand toolbar to access the other chapters and themes.

It is implicit that for many engineering topics, students have core competence in some mathematical topics such as polynomials, roots, complex numbers, exponentials, logarithms and Laplace.

This section contains the following topics. Under each topic there are hardcopy (pdf) notes, a video talk through of key derivations with example problems and also a tutorial sheet for users to test themselves.

  1. Simultaneous equations 1 - what is a straight line?
  2. Simultaneous equations 2 - equation of a straight line in everyday terms.
  3. Simultaneous equations 3 - algebraic formulae for a straight line.
  4. Simultaneous equations 4 - intercepts of straight lines.
  5. Simultaneous equations 5 - simple solution method.
  6. Simultaneous equations 6 - tutorial sheet.
  7. Simultaneous equations 7 - methods used in schools.
  8. Simultaneous equations 8 - introduction to advanced methods.
  9. Simultaneous equations 9 - matrix inverse.
  10. Simultaneous equations 10 - Cramer's rule.
  11. Simultaneous equations 11 - row echelon form.
  12. Simultaneous equations 12 - row operations.
  13. Simultaneous equations 13 - Gaussian elimination.
  14. Simultaneous equations 14 - matrix inverse with Gaussian elimination.

Simultaneous equations 1 - what is a straight line

Introduction to simple scenarios which can be described by a straight line equation.

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Simultaneous equations 2 - equation of a straight line in everyday terms

Introduces the concept of straight lines using everyday scenarios which are easy to relate to. Introduces concepts of gradient and intercept with the vertical axis, again in terms of everyday scenarios.

Quick test question

In the standard equation y = mx +c, which of the following is true?
A. m represents the intercept with the y-axis.
B. m represents the normal.
C. c represents the intercept with the y-axis.
D. None of the above.

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Simultaneous equations 3 - algebraic formulae for a straight line

Develops video 2 by introducing the concept of 'abstract' variables to represent the terms in a straight line and hence the general equation form of 'y=mx+c'.

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Simultaneous equations 4 - intercepts of straight lines

Use a number of everyday scenarios to explain the meaning and importance of simultaneous equations and the intercept point.

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Simultaneous equations 5 - simple solution method

Gives a simple everyday interpretation of simultaneous equations from which the video demonstrates a solution method which is simple and intuitive with minimal reliance on abstract mathematics.

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Simultaneous equations 6 - tutorial sheet

Gives a number of worked examples of solving simultaneous equations. Students can pause the video and try the examples themselves before watching the solutions.

Quick test question

A simple way of solving simultaneous equations is to:
A. Sketch both lines and find the intercept.
B. Subtract one equation from the other.
C. Keep guessing different pairs of values until you find some that work.
D. None of the above.

Simultaneous equations 7 - methods used in schools

This video introduces the algebraic methods used in school to solve simultaneous equations. It is shown that this method is in fact equivalent to he intutive method of earlier videos (and thus in fact could be avoided where students struggle with its abstract nature).

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Simultaneous equations 8 – introduction to advanced methods

This video introduces the concepts of simultaneous equations having a unique solution, no solution or an infinite number of solutions. It then links the existence of a unique solution scenario to the determinant of the variable coefficient matrix being non-zero. Note the two trivial typos at 12.25 and where a minus sign has not been carried forward into the B matrix.

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Simultaneous equations 9 – matrix inverse

It is possible to express simultaneous equations using an equivalent matrix-vector identity. Using this, the solution can be determined in simple analytic terms using matrix inverse. Such expressions are very useful for algebra, but in general finding the matrix inverse is an onerous task to be avoided if possible. This video presents the simple definition of a matrix inverse and indicates why such a method is inefficient in general.

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Simultaneous equations 10 – Cramer's rule

Cramer’s rule gives a simple algebraic definition of the solution for individual components in terms of determinants of the equation coefficients. This can be very convenient and efficient for low dimensional problems, but would not generally be used for large numbers of variables.

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Simultaneous equations 11 – row echelon form

A typical method for solving simultaneous equations involves successive elimination of different variables. This video shows that such a technique is equivalent to finding a representation in matrix form which involves an upper or lower triangular matrix or so called row-echelon form. Critically, once the equations are in row-echelon form, the solution through back substitution is very efficient.

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Simultaneous equations 12 – row operations

This resource looks at equivalence between matrix multiplication and row operations. The significance is that a row echelon form can be produced by multiplication on the left by a suitable matrix, and this matrix can be determined from an equivalent set of successive row operations. Such row operations are efficient and the back bone of practical and efficient simultaneous equation solvers.

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Simultaneous equations 13 – Gaussian elimination

Introduces Gaussian elimination methods for solving simultaneous equations and gives a number of worked examples to clarify the algorithm.

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Simultaneous equations 14 – matrix inverse with Gaussian elimination

Shows that the same Gaussian elimination methods used for solving simultaneous equations can, with just a minor augmentation, be used to find a matrix inverse in a computationally efficient manner. Gives a number of worked examples to clarify the algorithm.

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