You have worked with linear systems in the past.
Let's refresh what we know, and see how these systems get enhanced in Algebra.

definition
 A "system" of linear equations is a set of equations of straight lines.
Systems of equations are also know as "simultaneous equations".

When working with a "system" of equations, you are working with two or more equations at the same time. Our initial investigations worked with two linear equations at a time.
When a system deals with two linear equations, with each equation having two variables
to a power of 1, it is referred to as a 2 x 2 linear system.

2 x 2 linear system
 2 equations
2 unknowns
y = -2x + 5
y = x - 13
 Solution:
x = 6 and y = -7
or (6,-7)

When working with 2 x 2 linear systems, remember that the equations of lines
may be written in a variety of forms, such as:
     y = 2x + 1;     -5x + y = 7;     4x + 2y = -5;      y = 7
You may need to re-write the equations before beginning your solutions.

In Algebra, we will be expanding the difficulty level of the systems we investigate.

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Solving Linear Systems:
When solving a linear system, you will actually be solving for the point where the two lines intersect (where they cross one another). That specific point will lie on both lines, thus making both equations true.

There are three main methods used for solving systems of linear equations.
• Substitution Method: The goal of this algebraic method is to replace one of the equations with an equivalent expression by solving for one variable in one of the equations.
• Elimination Method: The goal of this algebraic method is to eliminate one of the variables using addition or subtraction of the equations. The remaining equation will easily yield either the x or the y coordinate of the intersection point.
• Graphical Method: The goal of this graphical method is to solve the system by graphing the lines either on graph paper, or on the graphing calculator, and locating the point of intersection.

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Solutions:
Keep in mind that the solution to a system of linear equation will be the location of the intersection of the two equations.

But, will there always be a solution?

The answer is "No". There are actually three possible cases of what may occur:
sysG1
One Solution
The slopes of the lines are different.
This is the most common situation.
sysG2
No Solution
The lines are parallel with the same slopes,
but different y-intercepts.
There is no intersection.
sysG3
Infinite Solutions
The lines are the SAME line (one on top of the other). The slopes are the same and the y-intercepts are the same. The lines intersect everywhere and ALL points on the line are solutions.

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Applications of Systems:
While we will be concentrating on the algebraic and graphical solutions to our 2 x 2 linear systems, remember that there are actual "real world" situations where these equations (and their solutions) can be applied. Note: the variables in our equations can be letters other than x and y.

example

Kyle's little sister, Jenny, claims that she can beat him in a snowmobile race to the next turn in the trail. Kyle takes the challenge and gives her a one minute head start. Jenny travels at 1640 feet per minute and Kyle travels at 2640 feet per minute. How long will it take Kyle to catch up with Jenny?
 BGrace

Solution:
Let d = distance in feet and t = time in minutes. (Remember distance = rate x time.)

Create TWO linear equations: BGequations1


Let's look at these equations graphically.
(for your calculator, x = t and y = d)

BGgraph
You can quickly see from the graph that Kyle catches up to Jenny somewhere between 2 and 3 minutes into Jenny's run. By using the "intersection" option on the calculator, we see that after Jenny is traveling t = 2.64 minutes, Kyle travels 4329.6 feet and catches her.
Since Kyle started one minute after Jenny, he traveled 2.64- 1 = 1.64 minutes before catching Jenny.

The next lessons will show how to solve these systems of equations.

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