In deterministic models good decisions bring about good outcomes. You get that what you expect; therefore, the outcome is deterministic i. This depends largely on how influential the uncontrollable factors are in determining the outcome of a decision, and how much information the decision-maker has in predicting these factors.

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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.A system of equations is a collection of two or more equations with the same set of unknowns.

In solving a system of equations, we try to find values for each of . In this tutorial we will be specifically looking at systems that have two equations and two unknowns.

Tutorial Solving Systems of Linear Equations in Three Variables will cover systems that have three equations and three unknowns. We will look at solving them three different ways: graphing, substitution method and elimination method. Now we have the 2 equations as shown below.

Notice that the \(j\) variable is just like the \(x\) variable and the \(d\) variable is just like the \(y\).

It’s easier to put in \(j\) and \(d\) so we can remember what they stand for when we get the answers. This is what we call a system, since we have to solve for more than one variable – we have to solve for 2 here. 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. Algebra 2 Here is a list of all of the skills students learn in Algebra 2! These skills are organized into categories, and you can move your mouse over any skill name to preview the skill.

Steps For Solving Real World Problems. Highlight the important information in the problem that will help write two equations. Define your variables.

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Algebra - Linear Systems with Two Variables