
How to solve simultaneous equations by substitution.
Theory:If two (or more) equations have the same variables and the same solutions then they are simultaneous equations.Â 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.Â 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’: xÂ
= 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 Â 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 Â (4) Substitute the known value of ‘y’ into the equation for ‘x’
derived in step 1: xÂ
= 3  y Substituting 2 for ‘y’: xÂ
= 3  2 Therefore: x =
1 The ‘Method’ section below shows you how to use Algematics to solve simultaneous equations by substitution. Â
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 1Â Enter the equations to solveClick 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. If
required, use the ‘ * ’ symbol for multiply, and the ‘ / ’ symbol for
divide. Â Â Â Â Â Â Â
Maths... Â
Â x
+ y = 3 Â Â Â Click the Â button, then type the second equation into the maths box: Â Â Â Â Â Â
Maths... Â
Â 2x
+ 3y = 8 Â Â Â and then click Step 2Â Isolate one variable in one of the equationsChoose 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.
Â 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... xÂ = 3  y Step 3Â Substitute for the isolated variable into the other equationClick 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.
Click Â (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 4Â Calculate the first variableNOTE:
Continuing from step 3, use the
methods explained in the article: “Solve equations with algebra”
. For the example, 2(3

y) + 3y = 8 Click Â (expand) to remove the brackets... 6

2y + 3y = 8 Click Â (simplify all): 6
+ y = 8 Subtract 6 from both sides:  Click on the input box and type:Â
 Click  Click Â (simplify) y = 2 Â Step 5Â Calculate the second variableClick on the equation that you derived in step 3. For
the example, click on the equation: xÂ
= 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.
Click Â (substitute) then Â (simplify all) to calculate the second variable. For the example, substituting will give you this: xÂ
= 3  2 and simplifying will give you this: xÂ = 1 Â
