The following system of equations has solution $(x,y)$. What is the value of $y$? $$\frac{1}{2}(2x+y)=\frac{21}{2}$$ $$y=2x$$

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When solving for $y$, we typically want to eliminate or substitute a value for $x$. Solving for $x$ in the second equation,

$$y=2x$$ $$\frac{1}{2}y=x$$

Substituting this value into the first equation,

$$\frac{1}{2}(2x+y)=\frac{21}{2}$$ $$\frac{1}{2}\Big(2\left(\frac{1}{2}y\right)+y\Big)=\frac{21}{2}$$ $$\frac{1}{2}(y+y)=\frac{21}{2}$$ $$\frac{1}{2}(2y)=\frac{21}{2}$$ $$2y=21$$ $$y=\boxed{\frac{21}{2}}$$

Like the previous question, we can make the equations more manageable before beginning the process of elimination or substitution.

$$\frac{1}{2}(2x+y)=\frac{21}{2}$$ $$\Big(2\Big)\Big( \frac{1}{2}(2x+y) \Big)=\frac{21}{2} \Big( 2 \Big)$$ $$2x+y=21$$