Math 030 - CooleyIntermediate Algebra OCC

Section 3.1 – Systems of EquationsIn Two Variables

A system of linear equations consists of two or more equations (in x and y) where a common solution is sought. The solution of this system will be an ordered pair (x, y) that satisfies all the equations in the system.

A system of equations is usually denoted by: , where the solution is written as an ordered pair. In this example, the ordered pair, (1, 6) satisfies both equations. Each equation corresponds to a line, when graphed, and the ordered pair, (1, 6), is the point of intersection of the two lines.

When a system of two linear equations is graphed,three physical situations (solutions) are possible:

Name of System: / Consistent , Independent / Inconsistent , Independent / Consistent , Dependent
Number of Solutions: / 1
(Exactly 1 solution) / 0
(No solution) / Infinitely many
(Infinite solutions)
What to look for: / Different slopes. / 1. Same slope.
2. Differenty-intercepts. / 1. Same slope.
2. Samey-intercepts.
What going on: / Non-parallel lines. / Parallel lines. / Same line.

There are five techniques for solving a system of equations that are discussed in our text:

  1. Graphically or Graphical Method. (see Section 3.1)
  2. Substitution Method. (see Section 3.2)
  3. Addition or Elimination Method. (see Section 3.2)
  4. Matrices. (see Section 3.6)
  5. Cramer’s Rule. (see Section 3.7)

Method: / Graphical / Substitution /

Addition/Elimination

Procedure: / Pictorial / Computational / Computational
Efficiency Rating: / Worst / Okay / Best
Explanation: / This method produces an approximate graphical solution. It is hard to get accurate results, but it does show students how the type of solution relates to the physical situation. / This method produces an exact solution. It is a medium difficulty computational technique. Generally, the Substitution Method involves fractions and is a little more time consuming than the Addition/Elimination Method. / This method produces an exact solution. It is also the quickest and most efficient method, and is preferred by the vast majority of students.

Consistent – A system of equations that has at least one solution.

Inconsistent – A system of equations that has no solution.

Independent – A system of equations with no more than one solution.

Dependent – A consistent system of equations that has infinitely many solutions..

 Examples:

Determine the solution to each system of equations graphically. If the system is dependent or inconsistent, state so.

a)b)c)

 Solution:

After putting each of the equations in slope–intercept form, we get…

Before we graph, examine the slopes and the y-intercepts. See if you can describe the nature of the lines (i.e., parallel, non-parallel, coincident). Then based off that information, how many solutions do you expect for each

system? Then, what type of solution do you think we have? (i.e., consistent, inconsistent, dependent, independent).

Lines: ______

# of Solutions:______

Type:______

Graph of solution (physical situation):

Algebraic Solution:

(4, 1)  Exactly one solution.No SolutionInfinite Solutions

Consistent & IndependentInconsistent & IndependentConsistent & Dependent

 Exercises:

Determine the solution to each system of equations graphically. State whether the system is consistent or inconsistent as well as dependent or independent.

1)

2)

1