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Start with the given system of equations:
Now in order to solve this system by using elimination/addition, we need to solve (or isolate) one variable. I'm going to solve for y.
In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for
, we would have to eliminate
(or vice versa).
So lets eliminate
. In order to do that, we need to have both
coefficients that are equal in magnitude but have opposite signs (for instance 2 and -2 are equal in magnitude but have opposite signs). This way they will add to zero. By adding to zero, they can be eliminated.
So to make the
coefficients equal in magnitude but opposite in sign, we need to multiply both
coefficients by some number to get them to an common number. So if we wanted to get
and
to some equal number, we could try to get them to the LCM.
Since the LCM of
and
is
, we need to multiply both sides of the top equation by
and multiply both sides of the bottom equation by
like this:
Multiply the top equation (both sides) by
Multiply the bottom equation (both sides) by
Distribute and multiply
Now add the equations together. In order to add 2 equations, group like terms and combine them
Combine like terms and simplify
Notice how the x terms cancel out
Simplify
Since this equation is
always true regardless of what x or y is, we have an infinte number of solutions. So this system is dependent.
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Answer:
a.) How many solutions does the system have?
The system has an infinite number of solutions.
b)
The solutions are simply points that lie on either line (since the two lines are identical)
(