# Problem of the Week Problem D and Solution Two Equations and Two Variables

## Problem

Two Equations and Two Variables

If $$2x=3y+11$$ and $$2^x=2^{4(y+1)}$$, determine the value of $$x+y$$.

## Solution

Solution 1

Since $$2^x=2^{4(y+1)}$$, it follows that $$x=4(y+1)$$, or $$x=4y+4$$. We now have the following two equations. \begin{align*} 2x&=3y+11 \tag{1}\\ x&=4y+4 \tag{2}\end{align*} We can substitute equation $$(2)$$ into equation $$(1)$$ for $$x$$. \begin{aligned} 2x&=3y+11\\ 2(4y+4)&=3y+11\\ 8y+8&=3y+11\\ 5y&=3\\ y&=\frac{3}{5}\end{aligned} Now, we can substitute $$y=\frac{3}{5}$$ into equation $$(2)$$ to solve for $$x$$. \begin{aligned} x&=4y+4\\ &=4\left(\frac{3}{5}\right)+4\\ &=\frac{12}{5}+\frac{20}{5}\\ &=\frac{32}{5}\end{aligned} Now that we have the values of $$x$$ and $$y$$, we can determine the value of $$x+y$$. $x+y=\frac{32}{5}+\frac{3}{5}=\frac{35}{5}=7$ Therefore, the value of $$x+y$$ is $$7$$.

Solution 2

We can solve this problem in a faster way without finding the values of $$x$$ and $$y$$. Since $$2^x=2^{4(y+1)}$$, it follows that $$x=4(y+1)$$, or $$x=4y+4$$. We now have the following two equations. \begin{align*} 2x&=3y+11 \tag{1}\\ x&=4y+4 \tag{2}\end{align*} We can subtract equation $$(2)$$ from equation $$(1)$$, and obtain the equation $$x=-y+7$$. Rearranging this equation gives $$x+y=7$$. Therefore, the value of $$x+y$$ is $$7$$.