Question Video: Solving Systems of Nonlinear Equations Mathematics

Given that 2^(π₯) β 2^(π¦) = 7 and 2^(π₯) + 2^(π¦) = 9, find the value of π₯ and the value of π¦.

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Video Transcript

Given that two to the π₯-power minus two to the π¦-power equals seven and two to the π₯-power plus two to the π¦-power equals nine, find the value of π₯ and the value of π¦.

What this question is telling us is we want the value of π₯ and π¦ that make both of these statements true at the same time; weβre looking for their solution. One way to find their solution would be to combine these two equations. When we do that, we get two to the π₯-power plus two to the π₯-power. And the simplest way to write that would be two times two to the π₯-power. After that, negative two to the π¦-power plus positive two to the π¦-power equals zero, and seven plus nine equals 16. Now we want to try and isolate π₯. We divide both sides of the equation by two, which tell us that two to the π₯-power is equal to eight.

Since we have a variable as our exponent, we want to try to rewrite eight so that itβs an exponent with a base two. We know that two cubed equals eight. If we substitute two cubed in place of eight, we can say two to the π₯-power is equal to two cubed, and therefore π₯ must be equal to three. To find our π¦-variable, weβll plug in what we know for π₯. And if weβve done this correctly, we can plug that π₯-value into both of these equations and solve for π¦, and the π¦-values will be the same for either equation. This means weβll have two cubed minus two to the π¦-power equals seven and two cubed plus two to the π¦-power equals nine. For the first equation, that is eight minus two to the π¦-power equals seven.

To isolate π¦, we subtract eight from both sides, and we see that negative two to the π¦-power equals negative one. Multiplying through by negative one, which means two to the π¦-power is equal to one. And this means that π¦ has to be equal to zero. Since we have a base of two, the only exponential value that will make two to the π¦-power equal to one is zero. We can check that this is true by solving our second equation. Again, weβll need to subtract eight from both sides, and we end up with two to the π¦-power equals one, which means, based on our exponents rules, that π¦ has to be equal to zero. Weβre saying that the values π₯ equals three and π¦ equals zero make both of these statements true.