# A blue spruce grows and average of 6 inches per year. A hemlock grows an average of 4 inches per year. If a blue spruce is 4 feet tall and a hemlock is 6 feet tall, when would u expect the trees to be the same height?

Jun 2, 2017

See a solution process below:

#### Explanation:

The expression for the height of the blue spruce would be:

$4 + \frac{1}{2} y$ Where 4 is the starting height in feet and $\frac{1}{2} y$ is how much it grows every year (6 inches = $\frac{1}{2}$ foot per year).

The expression for the height of the hemlock would be:

$6 + \frac{1}{3} y$Where 6 is the starting height in feet and $\frac{1}{3} y$ is how much it grows every year (4 inches = $\frac{1}{3}$ foot per year).

We can now equate these two expressions and solve for $y$:

$4 + \frac{1}{2} y = 6 + \frac{1}{3} y$

$- \textcolor{red}{4} + 4 + \frac{1}{2} y - \textcolor{red}{\frac{1}{3} y} = - \textcolor{red}{4} + 6 + \frac{1}{3} y - \textcolor{red}{\frac{1}{3} y}$

$0 + \frac{1}{2} y - \textcolor{red}{\frac{1}{3} y} = 2 + 0$

$\frac{1}{2} y - \frac{1}{3} y = 2$

$\textcolor{red}{6} \left(\frac{1}{2} y - \frac{1}{3} y\right) = \textcolor{red}{6} \times 2$

$\left(\textcolor{red}{6} \times \frac{1}{2} y\right) - \left(\textcolor{red}{6} \times \frac{1}{3} y\right) = 12$

$3 y - 2 y = 12$

$\left(3 - 2\right) y = 12$

$1 y = 12$

$y = 12$

It would be expected to take 12 years for both trees to be the same height.