Simultaneous Equations by Substitution
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How to solve simultaneous equations by substitution.

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

Theory:

If two (or more) equations have the same variables and the same solutions then they are simultaneous equations.  For example, these equations are simultaneous equations:

x + y = 3 and

2x + 3y = 8

because both have the same variables: ‘x’ and ‘y’, and the same solutions: x = 1, y = 2

Substituting x = 1 and y = 2 into both equations, they BOTH give correct answers:

1 + 2 = 3 and

2΄1 + 3΄2 = 8

Thus: x = 1 and y = 2 are the solutions to both equations.

'Solving' simultaneous equations means finding the values of 'x' and 'y' that make them true. The following steps will demonstrate how to solve simultaneous equations by the substitution method

We will use the example equations above to demonstrate the procedure...

(1) Isolate one of the variables ( ‘x’ ) on one side of one of the equations:

x + y = 3

Isolating ‘x’:

x  = 3 - y

(2) Substitute for the isolated variable in the other equation:

2x + 3y = 8

Substituting 3 - y for ‘x’:

2(3 - y) + 3y = 8

 This equation has only one variable, so we can solve it.

(3) Solve this equation for the other variable, ‘y’:

2(3 - y) + 3y = 8

Expanding the brackets:

6 - 2y + 3y = 8

Simplifying:

6 + y = 8

Subtracting 6 from both sides:

y = 2

 (4) Substitute the known value of ‘y’ into the equation for ‘x’ derived in step 1:

x  = 3 - y

Substituting 2 for ‘y’:

x  = 3 - 2

Therefore:

x = 1

The ‘Method’ section below shows you how to use Algematics to solve simultaneous equations by substitution.

 

Download the free support file... We have created an Algematics document containing the completed example from this topic. It also includes practice exercises to improve your skills.

File name:  'Simultaneous equations (substitution).alg'   File size: 6kb
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to download the file.

If you choose 'Open this file from its current location', then Algematics should open the document immediately. If not, try the other option: 'Save this file to disk', then run Algematics and choose the 'Open' command from the 'File' menu. Locate the saved file and open it. If you do not yet have Algematics installed on your computer, click here for instructions.

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'.

Step 1  Enter the equations to solve

Click  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  button until the message: ‘INSERT’ appears.

If required, use the ‘ * ’ symbol for multiply, and the ‘ / ’ symbol for divide.

 

       Maths...

   x + y = 3

 

 

 

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

       Maths...

   2x + 3y = 8

 

 

 

and then click

Step 2  Isolate one variable in one of the equations

Choose the equation in which you want to isolate a variable. (The other equation will be used in step 4 below.)

For the example, we choose the equation:

x + y = 3

Click on this equation with the mouse to make it the target data set.

Isolate one of the variables in this equation.

The procedure for isolating a variable is explained in the algebra topic:
       “ Solve equations with algebra”
.  

 

To isolate ‘x’ in the example, we subtract ‘y’ from both sides and then simplify. Type 'y' into the input box, then click the '-' toolbar button, the click the left hand 'S' toolbar button to simplify...

x  = 3 - y

Step 3  Substitute for the isolated variable into the other equation

Click on the other equation with the mouse to make it the target data set.

For the example, we click on the equation:

2x + 3y = 8

Click on the input box arrow and select the equation you derived in step 3.

For the example, select: ‘x = 3 – y’ in the input box.

Input 

   x = 3 - y

Click  (substitute) to substitute for the isolated variable.

You will now have an equation in only one variable that will be solved in step 5.

For the example, you will now have this equation:

2(3 - y) + 3y = 8

Step 4  Calculate the first variable

NOTE: Continuing from step 3, use the methods explained in the article: “Solve equations with algebra” .

For the example,

2(3 - y) + 3y = 8

Click  (expand) to remove the brackets...

6 - 2y + 3y = 8

Click  (simplify all):

6 + y = 8

Subtract 6 from both sides:

- Click on the input box and type: 

Input 

   6

- Click

- Click  (simplify)

y = 2

 

Step 5  Calculate the second variable

Click on the equation that you derived in step 3.

For the example, click on the equation:

x  = 3 - y

Click on the input box, and select or type the equation that you derived in step 4.

For the example, select: ‘x = 2’ in the input box.

Input 

   y = 2

Click  (substitute) then  (simplify all) to calculate the second variable.

For the example, substituting will give you this:

x  = 3 - 2

and simplifying will give you this:

x  = 1

 

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