Lesson HOW TO Solve Simultaneous Equations (systems of Linear Equations)

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<h3>1.6 The Basics</h3>

Any question or problem having one unknown variable requires a minimum of 1 equation. One usually should do the trick.

Likewise, any question or problem having two unknown variables requires a minimum of 2 equations. One equation is NOT enough to find both x and y, for example.

<b>In general, having n variables requires a minimum of n equations.</b>

Either you are given these or you will have to convert the English from a question into maths. So, a question asking you to find the age of a dad and his son, means there are 2 unknowns, so the question <b>MUST</b> hold enough information for you to get 2 equations.

There are 2 main methods of working through simultaneous equations: substitution and elimination. Each is one side of a coin. By that I mean both methods are really the same, just approached from a different viewpoint.

<h3>1.7 Substitution Method</h3>

EXAMPLE:
Solve 2x-y=6 and x+y=6. Note there are 2 variables (x and y) and we have 2 equations... so this is solveable.

Let 2x-y=6 be eqn1
Let  x+y=6 be eqn2

OK then... we will re-write one equation to say either x= or y=. Then we put this into the other equation, which then will just have 1 variable, so we can solve it.

Which equation and which variable do we choose? Very easy...the simplest looking to you! Why do more work than you have to?

Looking at the 2 equations, eqn2 looks the simpler: it has smaller coefficients (the numbers in front of the x and y). As to x or y? Here, they both look the same, so I will pick x.

so, eqn2 becomes x = 6-y. Call this eqn3.

We will now substitute eqn2 (written in the form of eqn3) into eqn 1 to get rid of x. So, eqn1 becomes...

2(6-y)-y = 6
12-2y-y = 6
12-3y = 6
-3y = -6
y = (-6)/(-3)
--> y = 2

We can now find x by putting this value into eqn2 (yes, you could put it into eqn3, but stick with the "original" versions, just in case you have made some sort of mistake somewhere.

Eqn2 becomes x+2 = 6, so x must be 4.

<h3>1.8 Elimination Method</h3>

EXAMPLE:
Solve 2x-y=6 and x+y=6.

Let 2x-y=6 be eqn1
Let  x+y=6 be eqn2

The elimination method lets you look at the 2 equations together and just by adding or subtracting them, we get rid of 1 variable, leaving the other to be found.

This method only works if the variable you are trying to get rid of has the same number in both equations.

In our example, the x terms are different (one has a 2 and the other has a 1). However, the y terms are the same, albeit different sign.

So, in my head, my thoughts are: "the y-terms have the same number" which means i can use elimination very simply here.

Next question: Do I add ot subtract them? Just remember the following:

<b>SSSS</b>ame sign --> <b>SSSS</b>ubtract
<b>DDDD</b>ifferent sign --> A<b>DDDD</b>

So, lets do it....

2x-y=6
 x+y=6

different signs therefore ADD! gives

3x = 12
--> x = 4

So now, pick one of your 2 equations to put this value into, to find y. Again, pick the simpler looking one, which here is eqn2.

eqn2 becomes 4+y = 6
so, y = 2

NOW CHECK by putting both values into the equation you have not used (here, eqn1)...

2(4)-2 is 8-2 which is 6, which is what eqn1 says it should equal. Now, we <b>know</b> our answers are correct.

<h3>1.9 Summary</h3>

Both methods are good and each can be used effectively to solve simultaneous equations. As to which is better? That is for you to think about.

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<b>EXAMPLES FROM THE WEBSITE</b>

This example is a common type where you have a 2 digit number that is reversed.
*[problems 26725/14541]