Video Transcript
If 𝑥𝑦 is equal to four and 𝑥
plus five 𝑦 is equal to negative three, what is the value of 𝑥 cubed plus 125𝑦
cubed?
In this question, we are asked to
determine the value of 𝑥 cubed plus 125𝑦 cubed using two equations involving 𝑥
and 𝑦. We might be tempted to try and
solve the equations for 𝑥 and 𝑦 or rewrite the expression in terms of a single
variable. However, there is an easier
method. If we look at the expression whose
value we want to find, we can note that it is the sum of two cubes. In particular, it is equal to 𝑥
cubed plus five 𝑦 cubed.
We can then recall that we can
factor a sum of cubes using the formula 𝑎 cubed plus 𝑏 cubed is equal to 𝑎 plus
𝑏 multiplied by 𝑎 squared minus 𝑎𝑏 plus 𝑏 squared. Substituting 𝑎 is equal to 𝑥 and
𝑏 is equal to five 𝑦 into the sum of cubes formula gives us 𝑥 plus five 𝑦
multiplied by 𝑥 squared minus 𝑥 times five 𝑦 plus five 𝑦 squared. We can then simplify the second
factor to obtain 𝑥 plus five 𝑦 multiplied by 𝑥 squared minus five 𝑥𝑦 plus 25𝑦
squared.
At this point, we know the value of
𝑥 plus five 𝑦. And we can find the value of
negative five 𝑥𝑦. However, we do not know the value
of 𝑥 squared plus 25𝑦 squared. We can find the value of this
expression by noting that it is very similar to the square of 𝑥 plus five 𝑦. Therefore, we will square both
sides of one of the given equations to get 𝑥 plus five 𝑦 all squared is equal to
negative three squared. We can then distribute the exponent
and evaluate to obtain 𝑥 squared plus 10𝑥𝑦 plus 25𝑦 squared is equal to
nine.
Remember, we are told that 𝑥𝑦 is
equal to four. We can substitute this value into
the equation to get 𝑥 squared plus 10 times four plus 25𝑦 squared is equal to
nine. We can then rearrange this equation
to see that 𝑥 squared plus 25𝑦 squared is equal to negative 31. We can now substitute all of these
values into our expression. This gives us negative three
multiplied by negative 31 minus five times four. Finally, we can evaluate this
expression to see that 𝑥 cubed plus 125𝑦 cubed is equal to 153.
