Question Video: Solving Systems of Exponential Equations Mathematics

Given that 2^(π‘₯) βˆ’ 𝑦 = βˆ’2 and 2^(π‘₯) + 𝑦 = 18, find the value of π‘₯ and the value of 𝑦.


Video Transcript

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

We’re trying to find the values of π‘₯ 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 π‘₯ power plus two to the π‘₯ power, we get two times two to the π‘₯ 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 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, we’ll take what we know about π‘₯, 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 π‘₯ correctly, then either of these equations should produce the same value for 𝑦 when we plug in three for π‘₯. 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 π‘₯ and 𝑦 that are true for both of these equations π‘₯ equals three, 𝑦 equals 10.

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