Math 030 - Cooleyintermediate Algebra OCC

Math 030 - CooleyIntermediate Algebra OCC

Chapter 3.1 – Solving Systems of Linear Equations by the Graphing Method

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

1. Graphically or Graphing Method. (see Chapter3.1)
2. Substitution Method. (see Chapter 3.2)
3. Addition or Elimination Method. (see Chapter 3.3)

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, coincide/coinciding). Then based off that information, how many solutions do you expect for eachsystem? Then, what type of solution do you think we have? (i.e., consistent, inconsistent, dependent, independent).

Example A / Example B / Example C
Lines: / Non-parallel / Parallel / Coinciding
# of Solutions: / 1 / 0 / Infinite Solutions
Type: / Consistent, Independent / Inconsistent, Independent / Consistent, Dependent

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