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Table of Contents:
Substitution
Elimination
Graphing
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Substitution:
Note: the first equation
is the same as
Start with the second equation
Plug in
Distribute.
Combine like terms on the left side.
Subtract
from both sides.
Combine like terms on the right side.
Divide both sides by
to isolate
.
Reduce. So this is the first part of the answer.
Go back to the first equation
Plug in
.
Multiply
and
to get
.
Combine like terms. This is the second part of the answer.
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Answer:
So the solutions are
and
Which forms the ordered pair (0,6)
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Elimination:
Start with the first equation
Subtract 6 from both sides
Subtract "y" from both sides
Start with the given system of equations:
Multiply the both sides of the first equation by 4.
Distribute and multiply.
So we have the new system of equations:
Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:
Group like terms.
Combine like terms. Notice how the y terms cancel out.
Simplify.
Divide both sides by
to isolate
.
Reduce.
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Now go back to the first equation.
Plug in
.
Multiply.
Remove any zero terms.
Divide both sides by
to isolate
.
Reduce.
So our answers are
and
.
Which form the ordered pair
)
. Note: this is the same answer as before.
This means that the system is consistent and independent.
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Graphing:
Start with the first equation
Rearrange the equation
Looking at
we can see that the equation is in slope-intercept form
where the slope is
and the y-intercept is
Since
this tells us that the y-intercept is
.Remember the y-intercept is the point where the graph intersects with the y-axis
So we have one point
Now since the slope is comprised of the "rise" over the "run" this means
Also, because the slope is
, this means:
which shows us that the rise is 2 and the run is 1. This means that to go from point to point, we can go up 2 and over 1
So starting at
, go up 2 units
and to the right 1 unit to get to the next point
)
Now draw a line through these points to graph
So this is the graph of
through the points
and
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Now move onto the second equation
Subtract
from both sides.
Rearrange the terms.
Divide both sides by
to isolate y.
Break up the fraction.
Reduce.
Looking at
we can see that the equation is in slope-intercept form
where the slope is
and the y-intercept is
Since
this tells us that the y-intercept is
.Remember the y-intercept is the point where the graph intersects with the y-axis
So we have one point
Now since the slope is comprised of the "rise" over the "run" this means
Also, because the slope is
, this means:
which shows us that the rise is -3 and the run is 4. This means that to go from point to point, we can go down 3 and over 4
So starting at
, go down 3 units
and to the right 4 units to get to the next point
)
Now draw a line through these points to graph
So this is the graph of
through the points
and
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Now let's graph the two equations together on the same coordinate system:
Graph of
(red) and graph of
(green)
Notice how the two lines intersect at the point (0,6). So this means that the solution is
and
(which confirms our previous answers)
