History[ edit ] Historically, equations of motion first appeared in classical mechanics to describe the motion of massive objectsa notable application was to celestial mechanics to predict the motion of the planets as if they orbit like clockwork this was how Neptune was predicted before its discoveryand also investigate the stability of the solar system.

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Linear Systems with Two Variables A linear system of two equations with two variables is any system that can be written in the form. Also, the system is called linear if the variables are only to the first power, are only in the numerator and there are no products of variables in any of the equations.

Here is an example of a system with numbers. This is easy enough to check. Do not worry about how we got these values. This will be the very first system that we solve when we get into examples. Note that it is important that the pair of numbers satisfy both equations. Now, just what does a solution to a system of two equations represent?

Well if you think about it both of the equations in the system are lines. As you can see the solution to the system is the coordinates of the point where the two lines intersect.

So, when solving linear systems with two variables we are really asking where the two lines will intersect. We will be looking at two methods for solving systems in this section.

The first method is called the method of substitution. In this method we will solve one of the equations for one of the variables and substitute this into the other equation.

This will yield one equation with one variable that we can solve. Once this is solved we substitute this value back into one of the equations to find the value of the remaining variable. In words this method is not always very clear.

Example 1 Solve each of the following systems. We already know the solution, but this will give us a chance to verify the values that we wrote down for the solution. Now, the method says that we need to solve one of the equations for one of the variables.

This means we should try to avoid fractions if at all possible.

This is one of the more common mistakes students make in solving systems. Here is that work. As with single equations we could always go back and check this solution by plugging it into both equations and making sure that it does satisfy both equations.

Note as well that we really would need to plug into both equations. It is quite possible that a mistake could result in a pair of numbers that would satisfy one of the equations but not the other one.

As we saw in the last part of the previous example the method of substitution will often force us to deal with fractions, which adds to the likelihood of mistakes. This second method will not have this problem. If fractions are going to show up they will only show up in the final step and they will only show up if the solution contains fractions.

This second method is called the method of elimination. In this method we multiply one or both of the equations by appropriate numbers i. Then next step is to add the two equations together.

Because one of the variables had the same coefficient with opposite signs it will be eliminated when we add the two equations. The result will be a single equation that we can solve for one of the variables. Once this is done substitute this answer back into one of the original equations. Example 2 Problem Statement.

Working it here will show the differences between the two methods and it will also show that either method can be used to get the solution to a system.

So, we need to multiply one or both equations by constants so that one of the variables has the same coefficient with opposite signs.draw two lines that could represent this system To solve the system of linear equations and by using the linear combination method, Henry decided that Consider the system of linear equations.

2y = x + 10 3y = 3x + 15 Which statements about the system are The point (0, 2) is the only solution to the system of linear equations that contains the. Algebra -> Coordinate Systems and Linear Equations -> SOLUTION: Write a system of two linear equations that has. a) only one solution,(5,1).

a) only one solution,(5,1). b) . Solve the equation. The examples done in this lesson will be linear equations. Solutions will be shown, but may not be as detailed as you would like. A System of Equations has two or more equations in one or more variables Many Variables So a System of Equations could have many equations and many variables.

Differential equations involve the differential of a quantity: how rapidly that quantity changes with respect to change in another. For instance, an ordinary differential equation in x(t) might involve x, t, dx/dt, d 2 x/dt 2 and perhaps other derivatives.

We'll look at two simple examples of. There is only solutions to the systems of equations - y = x -2 & y = -x + 2. We can find this by looking at the slopes of each line, which is 1 and

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Differential Equations: some simple examples from Physclips