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: