Systems of Linear Equations in Two Variables
A system of linear equations is just two or more equations that are graphed on the same
graph.
A solution to a system of equations is the point (or points) that are on both lines.
To determine if an ordered pair is a solution, plug the values of the coordinates into each
equation. If they work in both, then that point is on both lines and it is a solution.
When two straight lines are graphed on the same graph, one of three things will happen:
The lines intersect in a single point. That point of intersection is the solution of the
system of equations. We call this system consistent and we call the equations
independent.
The lines are coincident. That means that when we graph the second equation, it lies
right on top of the first line. These lines have an infinite number of points in common.
We call this system consistent and we call the equations dependent.
The lines are parallel. They have no point of intersection. There is no solution to this
system of equations. We call this system inconsistent.
Lines
intersect
coincident
parallel
System
consistent
consistent
inconsistent
Equations
independent
dependent
To solve a system of linear equations by graphing, we just graph them on the same graph
and look at them. If they intersect, the point of intersection is the solution. If we just have
one line, they are coincident. If they never meet, they are parallel.
To solve a system of linear equations by substitution:








solve one of the equations for a variable
“Bird’s nest” that variable in the second equation.
Fly in your bird (the complicated side of the first equation).
Solve.
Now “bird’s nest” your first equation for the variable you just found
Fly in your answer to get the second answer
When you are done, you should have a value for x and a value for y
That ordered pair (x, y) is the solution. [The point of intersection of the lines]
To solve a system of linear equations by elimination:






Rewrite both equations in standard form so that you have a column of x-terms, a
column of y-terms and a column of constants.
Multiply one of the equations so that either the x’s or the y’s in the two equations
have the same coefficient with the opposite sign.
Add the equations together. This should eliminate one variable.
Solve for the remaining variable.
Plug that answer into one of the original equations to get the second variable.
Plus both answers into the other equation to check your answers.
Regardless of which way you solve the system of equations (graphing, substitution or
elimination):
If the lines intersect: you will get a value for both x and y. (x, y) is the point of
intersection of the two lines.
If the lines are coincident: you will get an identity like 0 = 0
If the lines are parallel: you will get a contradiction like 3 = 0
NOTE: When the lines are coincident, the answer looks funny. The answer is all the
points on the line. Express the solution in set notation.
For example, if the equation is 2x + 4y = 8
The answer is { (x, y) | 2x + 4y = 8 } meaning “The set of all (x, y) such that 2x + 4y = 8”