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

In this case it looks like it will be really easy to solve the first equation for \(y\) so let’s do that.

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.

This is one of the more common mistakes students make in solving systems.

To so this we can either plug the \(x\) value into one of the original equations and solve for \(y\) or we can just plug it into our substitution that we found in the first step. \[y = 3x - 7 = 3\left( 2 \right) - 7 = - 1\] So, the solution is \(x = 2\) and \(y = - 1\) as we noted above.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.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. \[\begin3x - y & = 7\ 2x 3y & = 1\end\] Before we discuss how to solve systems we should first talk about just what a solution to a system of equations is.A solution to a system of equations is a value of \(x\) and a value of \(y\) that, when substituted into the equations, satisfies both equations at the same time.Click "Show Answer" underneath the problem to see the answer.Or click the "Show Answers" button at the bottom of the page to see all the answers at once.\[3x - 7 = y\] Now, substitute this into the second equation.\[2x 3\left( \right) = 1\] This is an equation in \(x\) that we can solve so let’s do that.For the example above \(x = 2\) and \(y = - 1\) is a solution to the system. \[\begin3\left( 2 \right) - \left( \right) & = 7\ 2\left( 2 \right) 3\left( \right) & = 1\end\] So, sure enough that pair of numbers is a solution to the system. 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.

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