A carpenter purchased 50 ft of redwood and 90 ft of pine for a total cost of $273. A second purchase, at the same prices, included 80 ft of redwood and 60 ft of pine for a total cost of $336. Find the cost per foot of redwood and of pine.
redwood	       dollars per foot
pine	       dollars per foot


To solve this problem, we can set up a system of linear equations based on the given information. Let's denote:

- $ r $ as the cost per foot of redwood (in dollars).
- $ p $ as the cost per foot of pine (in dollars).

### Given Information:
1. For the first purchase:
   \[
   50r + 90p = 273
   \]
2. For the second purchase:
   \[
   80r + 60p = 336
   \]

### Step 1: Set up the system of equations

We now have the following system of linear equations:

\[
\begin{align*}
50r + 90p & = 273 \quad \text{(Equation 1)} \\
80r + 60p & = 336 \quad \text{(Equation 2)}
\end{align*}
\]

### Step 2: Solve the system of equations

We can solve this system using the method of elimination or substitution. Let's use the elimination method.

#### Step 2.1: Multiply both equations to eliminate one variable

Multiply Equation 1 by 4 and Equation 2 by 5 to make the coefficients of $ p $ the same:

\[
\begin{align*}
4 \times (50r + 90p) & = 4 \times 273 \quad \Rightarrow \quad 200r + 360p = 1092 \quad \text{(Equation 3)} \\
5 \times (80r + 60p) & = 5 \times 336 \quad \Rightarrow \quad 400r + 300p = 1680 \quad \text{(Equation 4)}
\end{align*}
\]

#### Step 2.2: Subtract Equation 3 from Equation 4

\[
\begin{align*}
(400r + 300p) - (200r + 360p) & = 1680 - 1092 \\
200r - 60p & = 588
\end{align*}
\]

#### Step 2.3: Solve for $ r $

\[
200r = 588 + 60p \quad \Rightarrow \quad r = \frac{588 + 60p}{200}
\]

But to simplify, let's directly substitute the value from the subtraction:

\[
r = \frac{588}{200} = 2.94
\]

Thus, $ r = 2.94 $, the cost per foot of redwood.

### Step 3: Substitute $ r = 2.94 $ back into one of the original equations

Substitute $ r = 2.94 $ into Equation 1:

\[
50(2.94) + 90p = 273 \quad \Rightarrow \quad 147 + 90p = 273
\]

Solve for $ p $:

\[
90p = 273 - 147 = 126 \quad \Rightarrow \quad p = \frac{126}{90} = 1.4
\]

Thus, $ p = 1.4 $, the cost per foot of pine.

### Final Answer:
- **Redwood:** $2.94 per foot
- **Pine:** $1.4 per foot

Would you like more details or have any questions?

---

Here are 5 related questions to consider:
1. How would the solution change if the cost totals were different?
2. What if the quantities of redwood and pine were swapped in the first purchase?
3. Can you solve a similar problem using the substitution method?
4. How can this problem be visualized graphically?
5. What happens to the system of equations if one variable's price is fixed?

**Tip:** When solving systems of equations, always check your solution by substituting the values back into the original equations to ensure accuracy.

A carpenter purchased 50 ft of redwood and 90 ft of pine for a total cost of $273. A second purchase, at the same prices, included 80 ft of redwood and 60 ft of pine for a total cost of $336. Find the cost per foot of redwood and of pine.

Find the cost per foot of redwood and pine by solving a system of linear equations. A carpenter purchased 50 ft of redwood and 90 ft of pine for $273, and later 80 ft of redwood and 60 ft of pine for $336. Learn step-by-step algebraic methods for Grades 8-10.

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