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Linear Systems -Target B: Explain and Justify the Number of Solutions for any System of Linear Equations

 * [[image:Discuss this.png width="173" height="97"]] || Jeff thinks that two lines have to intersect each other on the coordinate plane. Deb disagrees with Jeff. Whom do you agree with and why? ||

Just like when we were solving single variable equations, when we solve a system of equations, there are different types of solutions that can exist: one solution, no solution, and infinitely many solutions. The solution to a system of equations is always the point(s) of intersection on the graph. If the two lines never intersect, then the system cannot have a solution. Let's review what it means for a system to have each of the types of solutions:

__One solution__: there is only one ordered pair, when substituted in the system for x and y, that will create a true statement for each equation __No Solution__: there is NO ordered pair that when substituted in to the system that will create a true statement for BOTH equations __Infinitely Many Solutions__: there are an INFINITE amount of ordered pairs that can be substituted in to the system that will create true statements for BOTH equations

If you were to graph a system with one solution, the graphs of the two lines would interesect at a one single point. The graphs of the lines of system with no solution would never intersect with each other, meaning that the two lines would be parallel to each other. If the system has infinitely many solutions, then the graphs of the two lines would fall on top of one another, intersecting at all points along the two lines. Another way to think of this is by thinking about snails. Yep, snails. A snail went across the sidewalk in a straight line and made a trail. Another snail went across the same sidewalk in the exact same place as the first snail and also made a trail. At how many locations are the snails trails the same?

Let's take a look at what these special types of solutions would look like graphically.

What about when solving a system using subsitution or linear combinations? What happens when you have a system that has no solution or infinitely many solutions? If a system of linear equations has no solution, all variable terms end up adding/subtracting out to zero, and you are left with a FALSE statement. If a system of linear equations has infinitely many solutions, all variable terms will add/subtract to zero, but you will be left with a TRUE statement. Look at the examples below. In example 1, since the system ends with a true statement, the system whould have infinitely many solutions. Since the system in example 2 ends with a false statement, this system would have no solution.


 * Linear Systems-Target B-Solution What Solution-Quick Check**

Solve the system to determine the number of solutions. math y = \text{-}4x - 5\ math math 8x + 2y = \text{-}10\ math

Linear Systems-Target B-Solution What Solution-Quick Check Solutions