A **system of equations** consists of more than one equation, considered together. Since we have been dealing with equations of lines, or **linear equations**, you may already suspect that the problems we look at today will consist of more than one line — in particular, we will restrict ourselves to systems involving two linear equations, and therefore two lines. A **solution **to a system of linear equations is a pair of numbers (x,y) that is a solution to *all* of the equations in our system — this means that when you substitute this particular x and y into *each* equation, it must make the equation true. It also means that, when checking to see if a pair of numbers is a solution, you *must* check the solution in each equation.

Imagine drawing two lines. Two lines will *usually* cross in exactly one point, and it is the coordinates (x,y) of this special point that give us the solution to the system of equations consisting of the two lines. This gives us one method for finding the solution to a system of linear equations — sketch the graphs of the two lines, and find the point where they cross. This is called the **graphical method**. (Note: the video should start about halfway through, at the beginning of a good example)

How can we find the solution without drawing the graph? There are several ways to find the solution using *algebra* — we will study one of them, the **substitution method**, today, and another, the **addition** or **elimination method**, in our next class.

In the **substitution method**, we solve an equation for one variable — this means getting one variable by itself — and then substitute the result into the other equation.

It is worth considering the different possibilities that can occur when we graph two lines. There are three:

- The two lines cross in exactly one point. In this case, there will be exactly one solution. (consistent and independent)
- The two lines do not cross at all. In this case, there will be no solutions. (inconsistent)
- The two lines, when graphed, turn out be exactly the same. In this case, they overlap completely — every point on one is also a point on the other. Here, there are infinitely many solutions. (consistent and dependent)

Finally, you may have noticed two pieces of terminology that were used above to describe these possibilities:

A system of equations is **consistent** if it has at least one solution, and it is **inconsistent** if it has no solutions. There are two types of consistent systems: those with a single solution are called **independent**, and those with infinitely many solutions are called **dependent**.

This video gives a good discussion of these ideas:

Professor Ezra

This video was so helpful.

today was a good class we reviewed most of what we learned

I feel that the Khan Academy Videos are way more helpful than the classroom, sadly. We dont understand anything that is taught for the majority of the time, and its out of hand

John

Prof. Reitz

systems of linear equations can get confusing but with the lessons and videos it can make life a lot easier.

Pro. Halleck

Was a bit difficult, some examples I understood others I was lost