WEBVTT

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Hi, my name is Richard.

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We're looking at Tamua 2021, paper two,
question 14.

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Here it is.

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Consider the following simultaneous
equations where P is a real number P * 2

a495a1ee-ad4c-4aa1-9252-8c79eb30eeea-1
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to the X plus log base 2 of y = 2 and two
to the X plus log base 2 of y = 1.

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The question is,
what is the complete range of P for which

65799642-89e3-41be-8b0d-daba73ef1e11-1
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these simultaneous equations have a real
solution XY?

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The first thing to point out is saying
that this has got a real solution.

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XY just means there's a pair of values
X&Y, both of which are real,

384e8a0c-09a5-4e7f-8897-4c0fdbbd9813-1
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which satisfy this.

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Something which jumps out immediately
about these two equations is that if P =

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1 here,
the two left hand sides would be exactly

b7c17c8c-71bb-4562-81d9-0a0aa1816cb1-2
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the same,
and the fact that the two right hand

b7c17c8c-71bb-4562-81d9-0a0aa1816cb1-3
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sides are different means that you would
not have a solution in that case.

03e1c4ea-f7b1-4c92-867c-d0e2d26e50ff-0
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So if P = 1,
these two equations are just not

03e1c4ea-f7b1-4c92-867c-d0e2d26e50ff-1
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consistent and they will not have a
solution.

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So we can kind of rule that case out
right from the start and we can work

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under the assumption that P is not equal
to 1.

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So let's think about how we might
approach solving these.

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We might notice that we have log base 2
of Y as a term on both of the left hand

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side.

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So it'll be very easy to eliminate that
term by subtracting one of the left hand

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side from the other.

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So let's call our equations 1 and 2 and
let's consider what we get when we do

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equation 1 minus equation 2.

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So we're going to subtract the left hand
side of two from the left hand side of

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one.

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We can see that these will cancel and
disappear and we will be left with P -,

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1 * 2 to the X.

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So the P2 to the X from the first
equation minus the two to the X from the

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other equation.

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Giving this on the right hand side,
we will have 2 -, 1, so we will have that.

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This equals 1.

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So we're trying to think about when there
will be a solution of this equation of

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real number X.

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We can see that this equation is the same
as saying that two to the X is 1 / P -, 1.

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Remember we've already ruled out the
possibility that P = 1 in this.

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We know that P = 1 will not give us any
solution, so we can do this.

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Now,
2 to the X is always a positive number.

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So if we think about the graph of y = 2
to the X,

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it's very much like the graph of in terms
of its shape and its position.

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It's like the graph of y = e to the X and
it's entirely above the X axis.

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So we can see that this equation will
only have a solution if the right hand

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side is positive.

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We would need that 1 / P - 1 to be a
positive value in order to find the value

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of X so that two to the X = P - 1.

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So from that we can see that X would
exist as a real value so long as P is

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bigger than one.

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Another way you might see that is is by
taking log base 2 of both sides.

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So if we take the log base two of two to
the X, that would just leave us back as X.

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And if we take the log base two of the
right hand side, we will have this.

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And once again,
we know that in order to apply log base 2

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to a value and thus get our value of X,
we need this value to be positive.

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So once again that leads us to P is
greater than one in order that that value

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is positive.

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OK,
so so far we can say that we will be able

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to find a real number X which satisfies
this so long as P is bigger than one.

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So let's imagine that we have found the X.

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Well,
the next thing we would naturally do in

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simultaneous equations is to go back and
try and find the Y which corresponds to

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that X.

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So we know that this X satisfies 2 to the
X = 1 / P - 1.

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So we could just substitute 1 / P - 1
here and then try to solve for Y.

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So what would that look like?

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We would have log base 2 of Y and
remember this is becoming 1 / P -,

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1 like here.

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So it comes over to the other side and we
have this.

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So to find our Y,
we need log base 2 of Y to be 1 -,

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1 / P - 1.

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But we can see that this Y will exist as
a real number just by thinking about the

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function log base 2.

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The graph of the function log base 2
looks like this and importantly it's

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image is the whole of the real line.

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So this goes down as far as we like.

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This part goes up as far as we like and
the image is the whole of the real line.

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So no matter which real number this is,
we will be able to find the real value of

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Y.

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So that log base 2 of Y is 1 -, 1 / P - 1.

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And so we need no further restrictions on
P to find Y.

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So in summary,
we saw earlier that to find the real X we

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need P to be bigger than one.

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Once we've found our real X,
there are no further restrictions on P to

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find a real Y.

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So all we need is that P is bigger than
one to have our real solution XY.

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And so our solution is C which P is
bigger than one.