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toc = Simultaneous Equations =


 * Simultaneous equations ** refer to when two (or more) equations are being considered at the same time.

To ** solve simultaneous equations ** means to find any points (x, y) where both the equations are true at the same time.

Solving Simultaneous Linear Equations
Recall that the graph of a function shows all the points (x, y) where that function is true.

If we graph two different linear functions the lines will meet at one point (unless they are parallel).

At the point where the two lines intersect, both functions will be true at the same time.

This means the coordinates of the point of intersection of the two graphs is the solution to the simultaneous equations.

Graphical Method
Some pairs of simultaneous equations can be solved by plotting the two graphs carefully on graph paper and observing their point of intersection.

**Example 1**
Solve simultaneously __**Solution:**__
 * y = 2x – 1 {and}
 * x + y = 5

Plot the two graphs carefully. (see right)

By observation of the graph: Solution is: ** (2, 3) **

Checking answer: Substitute ** (2, 3) ** into ** y = 2x – 1 **
 * 3 = 2 × 2 – 1 ü

Substitute ** (2, 3) ** into ** x + y = 5 **
 * 2 + 3 = 5 ü


 * (2, 3) ** makes both equations true, so ** (2, 3) ** is the correct solution.

Parallel Lines
Two linear functions produce ** parallel lines ** if they have the same __**gradient**__.

If we attempt to solve two simultaneous equations and the two lines are parallel then there is __**no solution**__.


 * Example 2 **

... ... ... y = 2x + 1 ... {and} ... ... ... y = 2x - 4


 * These two lines are __**parallel**__.
 * They have __**no**__ points of intersection
 * So there is ** no solution ** to these simultaneous equaitons.

Coincident Lines
If the two equations create exactly the same line, then we say they are ** coincident **.

If we attempt to solve two simultaneous equations and they both refer to the same line, then there is **__an infinite number of solutions__**. {ie a line is made up of an infinite number of points and every point that satisifies one equation also satisfies the other}


 * Example 3 **

... ... ... y = 2x – 1 ... ... {and} ... ... ... 2x – y = 1


 * These two lines are __**coincident**__
 * __**Every**__ point on the line is a point of intersection
 * There is an ** infinite number of solutions **.

Using a CAS calculator
All graphing software packages have a feature that reports the point of intersection of two graphs.

Like many graphing packages, the graphing part of Classpad needs the function to be entered in the form y = mx + c.

If the functions are not in that form (eg 2x + 4y = 20), go to the Main screen and enter: This make y the subject of the equation (ie provide it in the form y = mx + c)
 * **solve(2x + 4y = 20, y)**

Go to the Graphs & Tables screen.

Enter the two equations as **y1** and **y2**.

Click on the graph icon (1st icon on the left of the top bar).

{You may need to alter the ZOOM __or__ alter the WINDOW settings so the point of intersection is visible}

Go to ANALYSIS menu, G-SOLVE submenu and select __**intersect**__. {the coordinates of the point of intersection will appear at the bottom of the screen}

The ClassPad can also solve simultaneous equations in the Main Section (see here)

Using Graphmatica
{Graphmatica is a free download graphing program.}

Graphmatica does __not__ require functions to be in the form y = mx + c

Simply enter and graph the two equations as they are written.

Go to the TOOLS menu and select FIND INTERSECTION.

A pop-up box will appear
 * check that the two equations being shown are the ones you want to solve
 * click CALCULATE.

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