Given that two to the 𝑥-power minus two to the 𝑦-power equals seven and two to the 𝑥-power plus two to the 𝑦-power equals nine, find the value of 𝑥 and the value of 𝑦.
What this question is telling us is we want the value of 𝑥 and 𝑦 that make both of these statements true at the same time; we’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 𝑥-power plus two to the 𝑥-power. And the simplest way to write that would be two times two to the 𝑥-power. After that, negative two to the 𝑦-power plus positive two to the 𝑦-power equals zero, and seven plus nine equals 16. Now we want to try and isolate 𝑥. We divide both sides of the equation by two, which tell us that two to the 𝑥-power is equal to eight.
Since we have a variable as our exponent, we want to try to rewrite eight so that it’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 𝑥-power is equal to two cubed, and therefore 𝑥 must be equal to three. To find our 𝑦-variable, we’ll plug in what we know for 𝑥. And if we’ve done this correctly, we can plug that 𝑥-value into both of these equations and solve for 𝑦, and the 𝑦-values will be the same for either equation. This means we’ll have two cubed minus two to the 𝑦-power equals seven and two cubed plus two to the 𝑦-power equals nine. For the first equation, that is eight minus two to the 𝑦-power equals seven.
To isolate 𝑦, we subtract eight from both sides, and we see that negative two to the 𝑦-power equals negative one. Multiplying through by negative one, which means two to the 𝑦-power is equal to one. And this means that 𝑦 has to be equal to zero. Since we have a base of two, the only exponential value that will make two to the 𝑦-power equal to one is zero. We can check that this is true by solving our second equation. Again, we’ll need to subtract eight from both sides, and we end up with two to the 𝑦-power equals one, which means, based on our exponents rules, that 𝑦 has to be equal to zero. We’re saying that the values 𝑥 equals three and 𝑦 equals zero make both of these statements true.