# Chapter one

Mathematical skills

## Section nine: Simultaneous Equations

This chapter is on the theme of supporting mathematics that are needed for engineering problem solving, analysis, design and evaluation.

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 focusing on common sense approaches rather than abstract concepts.

### 1. What is a straight line?

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

A talk through video is on YouTube.

### 2. Equation of a straight line in everyday terms

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

A talk through video is on YouTube.

### 3. Algebraic formulae for a straight line

This develops from the previous video 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'.

A talk through video is on YouTube.

### 4. Intercepts of straight lines

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

A talk through video is on YouTube.

### 5. Simple solution method

This gives a simple everyday interpretation of simultaneous equations. The video demonstrates a solution method which is simple and intuitive with minimal reliance on abstract mathematics.

A talk through video is on YouTube.

### 6. Tutorial sheet

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

A talk through video is on YouTube.

### 7. Methods used in schools

This video introduces the algebraic methods used in schools to solve simultaneous equations. It is shown that this method is in fact equivalent to the intuitive method of earlier videos. It could be avoided where students struggle with its abstract nature.

A talk through video is on YouTube.

### 8. Advanced methods

This explains concepts of simultaneous equations having a unique solution, no solution or an infinite number of solutions. It links to the determinant of the coefficient matrix.

*There are two typing errors at 12 minutes 25 seconds, where a minus sign has not been carried forward into the B matrix.*

A talk through video is on YouTube. View the PowerPoint slides (PDF, 998KB).

### 9. Matrix inverse

Express simultaneous equations using an equivalent matrix-vector identity and solution using a matrix inverse. This is useful for algebra, but finding the matrix inverse is an onerous task to be avoided if possible.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 1.02MB).

### 10. Cramer's rule

Cramerâ€™s rule gives a simple algebraic definition of the solution for individual components, in terms of determinants of the equation coefficients. It is convenient and efficient for low dimensional problems, but would not generally be used for large numbers of variables.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 939KB).

### 11. Row echelon form

A typical method for solving simultaneous equations involves successive elimination of different variables. They have an equivalent in matrix form which involves an upper or lower triangular matrix. Once in row-echelon form, the solution through back substitution is very efficient.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 729KB).

### 12. Row operations

Row echelon form can be produced by multiplication on the left by a suitable matrix. This matrix can be determined from an equivalent set of successive row operations. Such row operations are the back bone of practical simultaneous equation solvers.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 1.12MB).

### 13. Gaussian elimination

This section introduces Gaussian elimination methods for solving simultaneous equations. It gives a number of worked examples to clarify the algorithm.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 970KB).

### 14. Matrix inverse with Gaussian elimination

This 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. There are a number of worked examples.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 931KB).