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How to solve simultaneous equations by elimination.

 This topic is part of the TCS FREE high school mathematics 'How-to Library', and will help you to solve simultaneous equations by elimination.Â Â Â  (See the index page for a list of all available topics in the library.)Â To make best use of this topic, you need to download the Algematics software. Click here for instructions.

### Theory:

In the ‘elimination’ method for solving simultaneous equations, two equations are simplified by adding them or subtracting them. This eliminates one of the variables so that the other variable can be found.

To add two equations, add the left hand expressions and right hand expressions separately. Similarly, to subtract two equations, subtract the left hand expressions from each other, and subtract the right hand expressions from each other. The following examples will make this clear.

Example 1:Â  Consider these equations:

2x - 5yÂ  =Â Â  1

3x + 5yÂ  = 14

The first equation contains a ‘-5y’ term, while the second equation contains a ‘+5y’ term. These two terms will cancel if added together, so we will add the equations to eliminate ‘y’.

To add the equations, add the left side expressions and the right side expressions separately.

 2x - 5yÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  = 1 +Â 3x + 5yÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  = + 14 (2x - 5y) + (3x + 5y)Â Â Â Â  = 1 + 14

Simplifying, -5y and +5y cancel out, so we have:

5xÂ Â  =Â Â  15

Therefore ‘x’ is 3.

By substituting 3 for ‘x’ into either of the two original equations we can find ‘y’.

Example 2:

The elimination method will only work if you can eliminate one of the variables by adding or subtracting the equations as in example 1 above. But for many simultaneous equations, this is not the case.

For example, consider these equations:

2x + 3yÂ  =Â Â  4

Â  x - 2yÂ  = -5

Adding or subtracting these equations will not cancel out the ‘x’ or ‘y’ terms.

Before using the elimination method you may have to multiply every term of one or both of the equations by some number so that equal terms can be eliminated.

We could eliminate ‘x’ for this example if the second equation had a ‘2x’ term instead of an ‘x’ term. By multiplying every term in the second equation by 2, the ‘x’ term will become ‘2x’, like this:

x´2 - 2y´2Â  = -5´2

giving:

2x - 4yÂ  = -10

Now the ‘x’ term in each equation is the same, and the equations can be subtracted to eliminate ‘x’:

 2x + 3yÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  = Â Â  4 -Â 2x - 4yÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  = -Â Â Â  -10 (2x + 3y) - (2x - 4y)Â Â Â Â  = 4 - -10

Removing the brackets and simplifying, the ‘2x’ terms cancel out, so we have:

7yÂ Â  =Â Â  14

So

y = 2

The other variable, ‘x’, can now be found by substituting 2 for ‘y’ into either of the original equations.

 Sometimes both equations must be modified in order to cancel a variable. For example, to cancel the ‘y’ terms for this example, we could multiply the first equation by 4, and the second equation by 3. Then there would be a ‘12y’ term in the first equation and a ‘-12y’ term in the second equation. Adding the equations would then eliminate ‘y’.

The steps in the ‘Method’ section below demonstrate how to use Algematics to solve simultaneous equations using the elimination method.

### Method:

IMPORTANT: This topic assumes that you know how to enter mathematical formulas into Algematics. Find out how by completing the three simple tutorials in the 'Getting Started' section of the Algematics program 'Help'.

To solve two equations by the elimination method:

-Â Â Â Â Â Â Â Â Â  enter the equations,

-Â Â Â Â Â Â Â Â Â  prepare them if necessary so that one of the variable will cancel,

-Â Â Â Â Â Â Â Â Â  add or subtract them to eliminate one of the variables, then

-Â Â Â Â  substitute to find the second variable.

We will use example 2 from the ‘Theory’ section above in this demonstration.

#### Â Step 1Â  Enter the equations to solve

Click Â and type your first equation into the maths box in the data entry dialog box.

If the ‘EMPTY’ message is not displayed between the blue buttons, click the Â button until the message: ‘INSERT’ appears.

Â Â Â Â Â Â Â Â Â Â Â  Maths...

Â  Â 2x + 3y = 4

Â

Â

Click the Â button, then type the second equation into the maths box:

Â Â Â Â Â Â Â Â Â Â Â  Maths...

Â  Â x – 2y = -5

Â

Â

Â Check the ‘Rule Of Line ...’ check box so that the two initial equations will stand out at the top of all the working.

Click

#### Step 2Â  If necessary, prepare either or both equations for eliminating a variable.

Decide which variable you are going to eliminate from the equations and whether you are going to add the equations or subtract them.

For the example equations,

2x + 3yÂ Â  =Â Â Â  4

Â  x - 2yÂ Â  =Â  -5

we will multiply the second equation through by 2 and eliminate variable ‘x’ by subtracting the second equation from the first.

To modify an equation, use a copy of the original equation.

To copy an equation, hold down the Ctrl key on the keyboard while you click on the equation with the mouse and drag it down below the others. The copy will appear when you release the mouse button.Â

For the example, copy the equation:

x - 2yÂ  =Â  -5

below the others, so that the display looks like this:
(The newly copied equation is shown here in red)

2x + 3yÂ  =Â Â  4

Â  x - 2yÂ  =Â  -5

Â  x - 2yÂ  =Â  -5

To multiply the bottom equation by 2:

Click on the input box and type ‘2’:Â Â

 InputÂ Â  2

Click

Multiplying by '2', both sides ...

(x - 2y)´2Â  = -5´2

Click Â (expand, this removes the brackets)

Expanding, both sides ...

x´2 - 2y´2Â  = -5´2

Click Â (simplify all)

Simplifying, both sides ...

2x - 4yÂ  = -10

RULE OFF: Double click the mouse on this last data set to open the data entry dialog box. Check the ‘Rule Off Line ...’ check box so that the two initial equations will stand out at the top of all the working.

Click

NOTE: If you need to modify the other equation as well, copy it to the bottom and follow the same procedure, ending with another rule off line.

#### Step 3Â  Eliminate the first variable

Copy the two equations that you are going to add or subtract to the bottom of the working page, using the copying technique described above.

For the example, the last few lines of the display will now look like this:

Simplifying, both sides ...

2x - 4yÂ  = -10

2x - 4yÂ  = -10

2x + 3yÂ  =Â Â  4

Â  Click on the bottom equation to make it the target.

 Â Â Â  2x + 3yÂ  =Â Â  4

Â

Click on the input box arrow and select the other equation.

For the example, select: ‘2x – 4y = -10’ in the input box.

 InputÂ Â  2x – 4y = -10

Click Â to add the equations, or Â to subtract them.
Â Â Â Â Â  (For the example, we are subtracting the equations to eliminate ‘2x’, so click Â )

Click Â (simplify all).Â Â  You will now have an equation in only one variable.

 For the example, the last few lines of the display will now look like this: 2x - 4yÂ  = -10 2x + 3yÂ  =Â Â  4 Subtracting '2x - 4y = -10' ... (2x + 3y) - (2x - 4y)Â Â  =Â Â  4 - -10 Simplifying, both sides ... 7yÂ  = 14Â  ï Solve this equation to find 'y'.

For the example, we just divide both sides by 7 and simplify. Click on the input box, type 7, then click , then . This gives the result:

yÂ  = 2

Make a rule off line under this last step as explained above.

#### Step 4Â  Substitute to find the other variable

Copy either of the original equations to the bottom of the working page, using the copying technique described earlier in this article.

For the example, we will copy the equation:

x - 2yÂ  = -5

Click on the input box, and type or select the equation for the known variable.

For the example, select or type: y = 2

 InputÂ Â  y = 2

Click Â (substitute) and then Â (simplify all).

You will now have an equation in the other variable. Solve this equation to find the value of this other variable.

For the example, the equation is:

x - 4Â  = -5

In this case, we add 4 to both sides and simplify:
Â Â Â Â Â Â  Click on the input box, type 4,click , then . This gives the result:

x Â =Â  -1

#### Step 5Â  [OPTIONAL] Validate the solutions by substitution

You can test your answers to make sure they are correct. Click on the input box, and type both solutions with a comma between them:

For the example, type:Â  x = -1, y = 2

 InputÂ Â  x = -1, y = 2

Click on one of the original equations to make it the target data set

2x + 3yÂ  =Â Â  4

Click Â (substitute)

2´-1 + 3´2Â  =Â Â  4

Click Â (simplify all)

4Â  =Â Â  4

Click on the other original equation to make it the target data set

x - 2yÂ Â  =Â  -5

Click Â (substitute)

-1 - 2´2Â  =Â Â  -5

Click Â (simplify all)

-5Â  =Â Â  -5

If you get equations that are true in both cases, then your solutions are correct !

For the example, we get these equations:

-5Â Â  =Â Â  -5

and

Â  4Â Â  =Â Â  4

so the solutions: x = -1,Â  y = 2 are correct.

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