Once you have expressions for hiking time and kayaking time, how can you combine them to represent the fact that she spent a total of 9 hours hiking and kayaking?

You are told she hiked 1.5 times as many miles as she kayaked. How can you write an equation connecting $h$ and $k$, and then use it to eliminate one variable from the time equation?

Let $x$ be the number of miles hiked and $y$ be the number of miles kayaked. From $h = 1.5k$, write $x = 1.5y$, which is the same as $y = \frac{2}{3}x$. From $\frac{h}{3} + \frac{k}{6} = 9$, substitute $x$ and $y$ to get $\frac{x}{3} + \frac{y}{6} = 9$, then multiply by 6 to get $2x + y = 54$, or $y = 54 - 2x$.

In Desmos, enter the two equations:

• y = (2/3)x
• y = 54 - 2x Then look for the point where the two lines intersect.

Click the intersection point of the two lines. The coordinates will appear as $(x, y)$. The $y$-value of this point represents the number of miles Priya kayaked during the weekend.

• Let $h$ be the number of miles Priya hiked and $k$ be the number of miles she kayaked.
• Time is distance divided by speed:
  • Hiking time: $h/3$ hours (because 3 miles per hour)
  • Kayaking time: $k/6$ hours (because 6 miles per hour)
• Total time is 9 hours, so:

• She hiked 1.5 times as many miles as she kayaked, so:

Now we have a system of two equations in $h$ and $k$. Next, we will use substitution to solve for $k$.

Use $h = 1.5k$ in the time equation:

Substitute $h$ with $1.5k$:

Write $1.5$ as the fraction $\frac{3}{2}$ to make the arithmetic clearer:

Divide $\frac{3}{2}k$ by 3:

So the equation becomes:

Now we just need to solve this equation for $k$.

Solve

Multiply both sides by 6 to clear denominators:

Divide both sides by 4:

So Priya kayaked 13.5 miles, which corresponds to answer choice C.