# Question Video: Solving Systems of Exponential Equations Mathematics

Given that 2^(π₯) β π¦ = β2 and 2^(π₯) + π¦ = 18, find the value of π₯ and the value of π¦.

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### Video Transcript

Given that two to the π₯ power minus π¦ is equal to negative two and two to the π₯ power plus π¦ is equal to 18, find the value of π₯ and the value of π¦.

Weβre trying to find the values of π₯ and π¦ 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 π₯ power plus two to the π₯ power, we get two times two to the π₯ power. And negative π¦ plus positive π¦ 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 π₯ 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 π₯ power is equal to two cubed, then π₯ must be equal to three. From there, weβll take what we know about π₯, plug it back into one of our equations to solve for π¦.

It doesnβt matter which of the equations we choose. If weβve solved for π₯ correctly, then either of these equations should produce the same value for π¦ when we plug in three for π₯. Two cubed equals eight. So, we have eight minus π¦ equals negative two. Subtracting eight from both sides gives us negative π¦ equals negative 10. And if we multiply through by negative one, we get π¦ equals 10. Now, letβs check our second equation. Two cubed equals eight. Eight plus π¦ equals 18. Then, we subtract eight from either side. And again, we see that π¦ equals 10.

The values for π₯ and π¦ that are true for both of these equations π₯ equals three, π¦ equals 10.