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A linear system of two equations with two variables is any system that can be written in the form.

\[\beginax by & = p\\ cx dy & = q\end\] where any of the constants can be zero with the exception that each equation must have at least one variable in it.

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.

\[3x - 7 = y\] 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.

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. Let’s work a couple of examples to see how this method works.

For instance, \(x = 1\) and \(y = - 4\) will satisfy the first equation, but not the second and so isn’t a solution to the system.

Likewise, \(x = - 1\) and \(y = 1\) will satisfy the second equation but not the first and so can’t be a solution to the system.

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