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Do you have to use the same variables for all equations when creating a system of equations?
The idea is to work with the same variables, but it is possible that some of the variables are missing in some of the equations.
What does it mean for a system of linear equations to have no solutions?
It means that there is no set of values for the variables such that all the linear equations are simultaneously true.
Can a system of linear equations in two variables have exactly two solutions?
Yes. The easiest case to see where this is true is in the case that the equations are all of degree = 1, which will yield one solution per variable.
What are applications for matrices?
Equations are an algebraic way of writing down a maths problem in shorthand. Two or more simultaneous equations may be used to describe the same problem. Matrices can be used to solve these simultaneous linear equations (that is equations with two or more unknown variables) and obtain the answer to those unknowns which satisfies both. Equations are therefore generally solved to get values of unknown variables....... Variable values are calculated (or assumed) to know all working or constant parameters of a system... e.g. for a chemical reaction; generally pressure, temperature, concentration of reactant etc., may be combinations of unknown variables. i.e. If these parameters are varied resultant yield get affected......... We never know all properties at start, we first found relations between variables by doing practicals & form equations......... Then these equations can be solved by many methods....... Out of these many methods matrices is one...... So which ever system can be represented by equations, matrices have application there........ e.g. engineering problems, weather forecasting, aerospace design, financial calculations, chemical processes, construction calculations etc........... And.....they were used by Albert Einstein to come up with his theories for General and Special Relativity.
How do you solve a linear system using substitution?
Suppose you have n linear equations in n unknown variables. Take any equation and rewrite it to make one of the variables the subject of the equation. That is, express that variable in terms of the other (n-1) variables. For example, x + 2y + 3z + 4w = 7 can be rewritten as x = 7 - 2y - 3z - 4w Then, in the other (n-1) equations, plug in that value for the variable and simplify (collect like terms). You will end up with (n-1) equations in (n-1) unknown variables. Repeat until you have only one equation in 1 variable. That gives you the value of one of the variables. Plug that value into one of the equations from the previous stage. These will be one of two equations in two variables. That will give you a second variable. Continue until you have all the variables. There are simpler methods using matrices but you need to have studied matrices before you can use those methods.
