Transcript 6-1a Solve System of Linear Equations

Solve Systems of
Linear Equations
Graphically
Honors Math – Grade 8
Get Ready for the Lesson
The equation y = 34.2 – 14.9x
represents the number of CD
singles sold (y) since the
number of years after 2000 (x).
The equation y = 3.3 + 4.7x
represents the number of music
videos (y) sold since the
number of years after 2000 (x).
These two equations can be
graphed on the same
coordinate plane.
The point at which the graphs of the two equations intersect represents the time
when the CD units sold equaled the music videos sold. The ordered pair of this
point is a solution of both equations.
Two equations such as y = 34.2 – 14.9x and y = 3.3 + 4.7x
together are called a system of equations.
A solution to a system
of equations is an
ordered pair that
satisfies both
equations. A system
can have no, one or an
infinite amount of
solutions.
If the graphs intersect or coincide, the system is said to be consistent.
If a consistent system has one solution it is said to be independent. If
it has infinite solutions it is dependent.
If the graphs are parallel, the system of equations is said to be
inconsistent.
Use the graph to determine whether each system has no
solution, one solution, or infinitely many solutions.
y  x  5
y  x 3
Since the graphs are intersecting,
there is one solution.
y  x  5
2 x  2 y  8
Since the graphs are parallel,
there is no solution.
Use the graph to determine whether each system has no
solution, one solution, or infinitely many solutions.
y  x  5
y  2 x  14
Since the graphs are intersecting,
there is one solution.
y  x  4
2 x  2 y  8
Since the graphs coincide, there
are infinitely many solutions.
Graph each system of equations. Then determine if the
system has one solution, no solution, or infinitely many
solutions. If the system has one solution, name it.
 y  x  8
 y  4x  7

Graph each
equation separately.
The graphs intersect at (3, 5).
Check to see if the solution satisfies
both equations.
y  x  8
5  (3)  8
55
Since the equation y = –x + 8 is in
slope intercept form. Use the slope
and the y-intercept to graph the line.
y  4x  7
5  4(3)  7
55
Since the equation y = 4x – 7 is in
slope intercept form. However, the
given graph does not have the
intercept of -7. Make a chart.
Graph each system of equations. Then determine if the
system has one solution, no solution, or infinitely many
solutions. If the system has one solution, name it.
x  2 y  6
2 x  4 y  4

Graph each
equation separately.
Remember the shortcut for standard form…
The slope is –A/B
The y-intercept is C/B
Graph the intercept
of 3 and use the
slope -1/2 to find
other points on the
line.
Graph the intercept
of 1 and use the
slope -1/2 to find
other points on the
line.
The graphs are parallel. There is
no solution.
Graph each system of equations. Then determine if the
system has one solution, no solution, or infinitely many
solutions. If the system has one solution, name it.
 y  2 x  3
2 x  y  3

Graph each
equation separately.
Remember the shortcut for standard form…
The slope is –A/B
The y-intercept is C/B
Notice that the slope
and the y-intercept
are the same as the
other line.
Since the equation y = –2x – 3 is in
slope intercept form. Use the slope
and the y-intercept to graph the line.
These lines coincide, therefore there are
infinitely many solutions.
Graph each system of equations. Then determine if the
system has one solution, no solution, or infinitely many
solutions. If the system has one solution, name it.
1

y   3 x  4

1
y  x  2
3

Graph each
equation separately.
Since the equation y = 1/3x + 2 is in
slope intercept form. Use the slope
and the y-intercept to graph the line.
Since the equation y = –1/3x + 4 is
in slope intercept form. Use the
slope and the y-intercept to graph
the line.
The lines intersect at (3, 3). Check
to see that this ordered pair satisfies
both equations in the system.