Systems of Equations
A system of equations is a set of two or more equations with the same variables. A
solution to a system of equations is a set of values for the variable that satisfy all the
equations simultaneously. In order to solve a system of equations, one must find all
the sets of values of the variables that constitutes solutions of the system.
In mathematics simultaneous equations are a set of equations containing multiple
variables. This set is often referred to as a system of equations. To solve
simultaneous equations, the solver needs to use the provided equations to find the
exact value of each variable. Generally the solver uses either a graphical method (by
plotting all the lines and/or curves on the same graph and finding the exact
coordinates of their intersection), the matrix method, the substitution method, and the
elimination method.
Systems of Linear Equations: Graphing
Thinking graphically, when you are solving systems, you are finding intersections. For
two-variable systems, there are three possible types of solutions:
The first graph shows two distinct non-parallel lines that cross at exactly one point.
This is called an "independent" system of equations, and the solution is always some
x,y-point.
The second graph shows two distinct lines that are parallel. Since parallel lines never
cross, then there can be no intersection; that is, for parallel lines, there can be no
solution. This is called an "inconsistent" system of equations, and it has no solution.
The third graph appears to show only one line. Actually, it's the same line drawn
twice. These "two" lines, really being the same line, then "intersect" at every point
along their length. This is called a "dependent" system, and the "solution" is the
whole line.
Solving Systems of Linear Equations Using Addition and
Subtraction
One disadvantage to solving systems using substitution is that isolating a variable
often involves dealing with messy fractions. There is another method for solving
systems of equations: the addition/subtraction method.
In the addition/subtraction method, the two equations in the system are added or
subtracted to create a new equation with only one variable. In order for the new
equation to have only one variable, the other variable must cancel out. In other
words, we must first perform operations on each equation until one term has an equal
and opposite coefficient as the corresponding term in the other equation.
We can produce equal and opposite coefficients simply by multiplying each equation
by an integer. Observe:
Example 1: Add and subtract to create a new equation with only one variable:
2 x  4 y  4

 x  3 y  13
Here, we can multiply the second equation by -2:
2 x  4 y  4

 2 x  6 y  26
Adding these two equations yields -2y = - 22
y=11
 y  11

 x  13  3 y
 x  20

 y  11