MAT017
Applied Math II
Procedures
To Solve a System of Linear Equations in 2 Variables
When we have 2 linear equations in 2 variables, there are 3 ways to solve the system.
1.
By Graphing
We investigated several methods of graphing in Chapter 8. When we graph 2 lines on the
same plane, we will determine the solution of the system, usually an ordered pair for the
point where the 2 lines intersect.
If the lines are parallel, there will be no solution, since the lines will never meet.
If the equations are equivalent, the lines will coincide, and all points on the line will be
solutions.
2.
By Addition
A second method we can use is solving by the Addition Method, which sometimes requires
Multiplication, as well.
The object is two have either the x or the y coefficients to be additive inverses of each other
so that when we add the 2 equations together one variable is eliminated. The steps for this
process are:
A. Multiply, if necessary, one or both equations(s) by a number that will
result in the coefficients of one variable being additive inverses.
(Example 3X and -3X).
B. Add the 2 equations together, resulting in a single equation in 1
variable.
C. Solve for the variable.
D. Plug the variable into one of the 2 original equations and solve for the
remaining unknown.
It is possible that in Step B, the result is other than an equation. Remember, we discussed the
fact that the 2 lines may be parallel or may coincide. In these cases, Steps C & D will not be
performed.
If in Step B, you arrive at a false statement like 0 = 2, the lines are parallel.
If in Step B, you arrive at a true statement like 0 = 0, the lines coincide.
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MAT017
Applied Math II
Procedures
To Solve a System of Linear Equations in 2 Variables
continued
3.
By Substitution
Another method for solving a system is the Substitution Method.
The object is to substitute an expression in terms of one variable into one of the equations.
The steps for this process are:
A.
B.
C.
D.
E.
Solve one of the equations for x or y, unless one already is in that
form.
Substitute the expression equal to that variable (x or y) into the
other equation.
Solve the resulting equation for the remaining variable.
Plug the value from C into the other equation and solve for the 2nd
variable.
You now have an ordered pair (x, y) that is the solution for the
system.
It is possible that in Step C, the result is other than a solution. Remember, we discussed the
fact that the 2 lines may be parallel or may coincide. In these cases, Steps D & E will not be
performed.
If in Step C, you arrive at a false statement like 0 = 2, the lines are parallel.
If in Step C, you arrive at a true statement like 0 = 0, the lines coincide.
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