Given that two to the 𝑥 power minus 𝑦 is equal to negative two and two to the 𝑥 power plus 𝑦 is equal to 18, find the value of 𝑥 and the value of 𝑦.
We’re trying to find the values of 𝑥 and 𝑦 that make both of these statements true, which means we want to solve them like a system of equations. One way to do that is to combine these two equations. If we add two to the 𝑥 power plus two to the 𝑥 power, we get two times two to the 𝑥 power. And negative 𝑦 plus positive 𝑦 equals zero. Negative two plus 18 equals 16. From there, we can divide both sides of the equation by two, which leaves us with two to the 𝑥 power equals eight.
To solve this further, we want to rewrite eight as an exponent with a base two. We know that eight is equal to two cubed. And if two to the 𝑥 power is equal to two cubed, then 𝑥 must be equal to three. From there, we’ll take what we know about 𝑥, plug it back into one of our equations to solve for 𝑦.
It doesn’t matter which of the equations we choose. If we’ve solved for 𝑥 correctly, then either of these equations should produce the same value for 𝑦 when we plug in three for 𝑥. Two cubed equals eight. So, we have eight minus 𝑦 equals negative two. Subtracting eight from both sides gives us negative 𝑦 equals negative 10. And if we multiply through by negative one, we get 𝑦 equals 10. Now, let’s check our second equation. Two cubed equals eight. Eight plus 𝑦 equals 18. Then, we subtract eight from either side. And again, we see that 𝑦 equals 10.
The values for 𝑥 and 𝑦 that are true for both of these equations 𝑥 equals three, 𝑦 equals 10.