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A linear-linear system of equations is a set of two or more equations which describe lines.

You can find the solution to a linear system of equations by finding all the ordered pairs (x, y) that satisfy each equation in the system. In other words, you want to find the intersection points of the graphs of the equations in the system.

You can estimate the solutions to a linear system by graphing each equation and finding the point(s) of intersection.

Let's look at an example to see how to solve a linear system of equations. Note that you will only work with linear systems consisting of two equations.

EXAMPLE 1:
Solve the following linear system of equations by graphing: 1. 2x + 3y - 18 = 0 2. x = y + 5
The equation 2x + 3y - 18 = 0 defines a line with y-intercept at 6 and a slope of -2/3. The equation x = y + 5 defines a line with y-intercept at -5 and a slope of 1.
After sketching the graph of both equations, you can estimate where they intersect. This is done for you in Figure 1 - zoom in to find a close estimate of the point of intersection.	FIGURE 1
It appears that the point of intersection is approximatly (6.6, 1.6).

It is often not possible to determine the exact solutions of a system of equations just by looing at the graph.

In order to find exact solutions, you must solve the system algebraically.

There are several algebraic strategies which can be used to solve systems of equations.

These strategies include the comparison, elimination, and substitution methods. Click to review how these strategies are used to solve linear-linear systems of equations.

In Example 1, a good strategy (but not the only strategy) for solving the system would be the substitution method:

Notice that the above linear-linear system of equations in Example 1 had one point of intersection, thus 1 solution.

In what other ways can linear-linear systems of equations intersect?

Good question! Besides intersecting at one point, there are only two other possibilities for how two lines can relate - the lines can be parallel, or they can coinicide, that is lie one on top of the other.

Since parallel lines do not intersect, this type of linear-linear systems has no solutions. Parallel lines are easy to spot because both lines will have the same slope.

Since lines that coincide are exactly the same line, they intersect at infinite many points and therefore have infinitely many solutions. It is also easy to spot these situations because lines that coincide either have exactly the same equation, or the equation of one line is a constant multiple of the equation of the other.

In summary, linear-linear systems of equations can have 0, 1 or infinitely many solutions!