Solutions of Systems of Equations

Jumat, 14 Juli 2023

Graphs of systems of linear equations may be intersecting lines, parallel lines, or the same line. Systems of equations can be described by the number of solutions they have.

The different possibilities for the graphs of two linear equations are summarized in the following table.

Example

State whether each system is consistent and independent, consistent and dependent, or inconsistent.

 

Alternative Solutions:

 

The graphs appear to be parallel lines. Since they do not intersect, there is no solution.

This system is inconsistent.

 

 

Alternative Solutions:

 

The graphs appear to intersect at the point at (–3, 5). Because there is one solution, this system of equations is consistent and independent.

 

Alternative Solutions:

 

Both equations have the same graph. Because any ordered pair on the graph will satisfy both equations, there are infinitely many solutions. The system is consistent and dependent.

 

You can determine the number of solutions to a system of equations by graphing.

 

Example

Determine whether each system of equations has one solution, no solution, or infinitely many solutions by graphing. If the system has one solution, name it.

 

4.    y = x + 2

y = –3x – 6

 

Alternative Solutions:

 

The graphs appear to intersect at (–2, 0).

Therefore, this system of equations has one solution, (–2, 0). Check that (–2, 0) is a solution to each equation.

 

Check:

The solution of the system of equations is (–2, 0).

 

5.    2x + y = 4

2x + y = 6

 

Alternative Solutions:

 

Write each equation in slope-intercept form.

 

The graphs have the same slope and different y-intercepts. The system of equations has no solution.

 

Example

Transportation Link

 

6.   The system of equations below represents the tracks of two trains. Do the tracks intersect, run parallel, or are the trains running on the same track? Explain.

 

Alternative Solutions:

 

x + 2y = 4

3x + 6y = 12

One equation is a multiple of the other. Each equation has the same graph and there are infinitely many solutions. Therefore, the trains are running on the same track.

 

Sumber

Labels: Mathematician

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