Mathematics Substitution 1 When you have a certain number of valid equations for a problem, with the same number of unknown variables in them, it is often possible to find a value for all of the unknown variables. We will start with a difficult example to show you the power of the method of substitution. What if you knew that with both equations being valid at the same time? What are the values of x and y ? If it is possible to solve such a problem, then a method called substitution will ALWAYS give us the answer.

Mathematics Substitution 2 When you have a certain number of valid equations for a problem, with the same number of unknown variables in them, it is often possible to find a value for all of the unknown variables. Solve one of the equations for one of the variables. In this case, we can solve the second equation for y. Substitute this variable into another equation. In this case, the first equation is the only one left.

Mathematics Substitution 3 When you have a certain number of valid equations for a problem, with the same number of unknown variables in them, it is often possible to find a value for all of the unknown variables. Then solve this equation (and any other remaining equation) for the variables that remain. Using a well known trigonometric identity, If the same function is found more than once in an equation, we can substitute for the function. (Yes. This is a different kind of substitution, but we need to know it as well.) Let’s set

Mathematics Substitution 4 When you have a certain number of valid equations for a problem, with the same number of unknown variables in them, it is often possible to find a value for all of the unknown variables. This is now a quadratic equation. Solve it for u. Reorganizing first

Mathematics Substitution 5 When you have a certain number of valid equations for a problem, with the same number of unknown variables in them, it is often possible to find a value for all of the unknown variables. We can now do some backtracking to find x and y. Only one of the answers (0.108) is reasonable, which usually happens in a real problem. Solving this last equation then gives us You should always use radians for angles unless you are told otherwise.

Mathematics Substitution 6 When you have a certain number of valid equations for a problem, with the same number of unknown variables in them, it is often possible to find a value for all of the unknown variables. We now have the value of one of the variables. We can therefore use either of the starting equations to fin the other variable. Finally, we can use the other equation to check our work as expected.

Mathematics Substitution 7 When you have a certain number of valid equations for a problem, with the same number of unknown variables in them, it is often possible to find a value for all of the unknown variables. Typically, you will see problems like this. What if you knew that with both equations being valid at the same time? What are the values of x and y? Could you solve this one?

Mathematics Substitution 8 There is another method that works in many cases and is easier when you have a lot of “simultaneous” equations Often, you will see problems like this. What if you knew that for all values of x, y and z? What are the values of x, y and z? You should be able to solve this with substitution, but we will now learn a new method.

Mathematics Substitution 9 There is another method that works in many cases and is easier when you have a lot of “simultaneous” equations “Simultaneous” equations are those that have the same function of x, the same function of y and the same function of z in them. These equations are simultaneous, because they all contain x, sin y and z, only. So, how do we solve this?

Mathematics Substitution 10 There is another method that works in many cases and is easier when you have a lot of “simultaneous” equations First, line up the equations like this and then number them. (1) (2) (3)

Mathematics Substitution 11 There is another method that works in many cases and is easier when you have a lot of “simultaneous” equations Then pick any two equations. Let’s use (1) and (3). We can multiply the same number on each side of any equation and still not change it. Let’s do this with both equations. (1) (2) (3) (1) (3)

Mathematics Substitution 12 There is another method that works in many cases and is easier when you have a lot of “simultaneous” equations They then become Notice, that by the right choice of multiplication the constants in front of z are opposite in these equations. This choice was on purpose. (1) (3) (1) (3)

Mathematics Substitution 13 There is another method that works in many cases and is easier when you have a lot of “simultaneous” equations We can now add these two equations together, to get If we do the same thing with two other equations, say (2) and (3), we would get (1) (3) (2) (3) (4) (5)

Mathematics Substitution 14 There is another method that works in many cases and is easier when you have a lot of “simultaneous” equations We are now down to two equations and two unknowns which we can solve the same way. (4) (5) (4) (5) (6)

Mathematics Substitution 15 There is another method that works in many cases and is easier when you have a lot of “simultaneous” equations Using our solution for y in either equation (4) or (5), we get and using these in either of equations (1), (2) or (3), we get

Mathematics Substitution 16 There is another method that works in many cases and is easier when you have a lot of “simultaneous” equations Now try another one yourself. What if you knew that for all values of x, y and z? What are the values of x, y and z? You should be able to do this using either the simultaneous equation method or the method of substitution.