Simultaneous equations refer to a set of equations. For example, the equations:

3x² - 2x + 3y = y - 1

2y - 3y² = 3x + 4   

are simultaneous equations with two unknowns.

                                            x +  y +   z = 4

The set of equations:   3x - 2y -   z = 4    

                                            x + 3y - 5z = -1  are three simultaneous equations with three unknowns.

The degree of simultaneous equations is given by the biggest power that any of the unknowns is raised to.

The example above shows two quadratic simultaneous equations with two unknowns.

The simultaneous equations are of first degree (linear) with two unknowns.

When the simultaneous equations are linear and non of the terms include unknowns which are multiplied together (x.y type) they are referred to as linear simultaneous equations.

In this unit we shall be working primarily with simultaneous equations of this type with two unknowns.

Graphical solution

Let's imagine that we want to solve the simultaneous equations:

By giving one of the unknowns a value and find the corresponding value of the other in each equation, we get a set of points (x, y) for each equation, which can be represented as straight lines on the graph.

In this example, we get:

The points (0 , 0) ; (1 , -2) for the first equation 2x + y = 0, which are enough to draw a straight line.

The points (0 , -1) , (3 , -4) for the second equation x + y = -1, which are used to draw the second straight line.

Look at these two lines in the following window.

2.- At which point is the solution found?

3.- What are the values of x and y at this point? Substitute these x and y values into the equations and check that they are satisfied.

The solution is shown in the following window:

1.- Note that the two lines cross at the point (1,-2).

"This means that the solution of the simultaneous equations is x = 1, y = -2".

2.- Check that if we substitute 1 for x and -2 for y in the simultaneous equations we get:

2·1-2 = 0; 2-2 = 0, "true" for the first equation

1-2 = -1, "true" for the second.

We have just found the graphical solution of the simultaneous equations.

 Solve the following simultaneous equations graphically:

1.-Change the equations written in the boxes in red (written correctly) and blue (which can be changed to the original equation).

2.-To find the solution change the values of x and y until the green lines cross at the point where the solution is found.

You should find that the solution is: x = 0.5 (-3/2) and y = 3.

You may have already worked on solving simultaneous equations in class and found that there are different methods which can be used to do so. For example, the simultaneous equations we solved in exercise 1:

can be solved easily using any of the methods:

By substitution:

- 1 - get one of the unknowns on its own on one side of the equation, for example in the first equation: y = -2x

- 2 - Substitute this value into the second equation: x - 2x = -1

- 3 - solve this equation: -x = -1 ; x = 1

- 4 - use this value to find the other unknown (step 1): y = -2

This solution is the same as the graphical solution we found earlier.

By elimination:

-1 - This is done by adding or subtracting the elements in both equations until an unknown can be eliminated. In order to do so the equations are simplified as much as possible and, if necessary, one of the equations is multiplied by a certain number. Thus, in this case one of the equations can be subtracted from the other and y can be eliminated: 1st - 2nd : x = 1

- 2 - The equation you are left with is solved. In this case it is already solved as the solution x is given directly: x = 1

- 3 - This solution is substituted into one of the two equations and the other unknown is found by solving the equation. In this case, by substituting x=1 into either of the two equations easily gives the solution y = -2. 

(The third of the most commonly known methods, called "equalization" will not be explained here as we consider the other two methods to be sufficient).

Exercise 3.- Solve the simultaneous equations from exercise 2 numerically in your exercise book. Choose the most appropriate method to do so and check that the solution is the same as the graphical one you found earlier.

Choose different pairs of simultaneous equations from your book and solve them numerically using one of the two methods explained above. Then check your solution by comparing it to the graphical solution given in the window of the section above, by first writing the equations into the boxes in the window.

Solve the following pair of simultaneous equations in your exercise book:

You will have probably got the equation 0 = -8 or something similar. What does this mean? It is obviously not true.

Look carefully at the window below: The two straight lines are parallel! Thus, they do not cross at any point. This means that this pair of simultaneous equations:

do not have a solution. 

Solve the following pair of simultaneous equations in your exercise book:

This time you should have obtained the expression 0 = 0 or another number = the same number. What does this mean?

The equality is true but both x and y have been eliminated. So what is the solution?

If the equality is true then will it be for any value of x or y?

Look at this window:

infinite solutions (there are infinite points on the line).

2.-Use the arrows at the bottom of the window to show this, changing the value of x. The point will move along the line and all the values given for x and y are solutions.

3.-Note that the solutions can be found numerically by assigning values for x or y in either of the two equations (they are both the same) and obtaining the corresponding value of the other unknown. For example, in the first equation:

x - 3 = y + 1, when y = 0 we get x = 4; when y = 2, x = 6; when y = -3, x = 1; etc. These are all solutions.

Many of the problems which are solved using equations may include more than one unknown and therefore involve the use of simultaneous equations. For example:

The problem: Find two numbers given that half of their sum is 5 and double their difference is 8.

The approach: Numbers: x and y.

The equations: (x + y) / 2 = 5 ; 2(x - y) = 8

Change the values of x and y in the following window to find the solution. Look carefully at the green lines until you find the point you need. Choose other problems to solve from your textbook. Form the equations and find the solutions by changing the value of the equations in the window above.

Quadratic and higher degree simultaneous equations (the latter being highly uncommon) can be solved graphically in the same way as with linear equations, and numerically using the same methods given to solve linear equations.

1.- Solve the simultaneous equations: x² + y² = 5 ; x - y = 1. 

2.-Note that the points of intersection of both graphs are (2, 1) and (-1, -2), which are the corresponding solutions you should have found for x and y.

3.- Find some pairs of quadratic simultaneous equations in your text book and solve them in your exercise book. Compare your solution to the graphical solution given in the window, by changing the equations written in the boxes accordingly.