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.
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.
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
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