A carpenter purchased 50 ft of redwood and 90 ft of pine for a total cost of $286. A second purchase, at the same prices, included 100 ft of redwood and 70 ft of pine for a total cost of $418. Find the cost per foot of redwood and of pine.


redwood

      dollars per foot

pine

      dollars per foot

We can solve this problem using a system of linear equations. Let:

- $ x $ represent the cost per foot of redwood (in dollars),
- $ y $ represent the cost per foot of pine (in dollars).

### Step 1: Set up the equations based on the problem.

From the first purchase, we have the equation:
\[
50x + 90y = 286
\]

From the second purchase, we have the equation:
\[
100x + 70y = 418
\]

### Step 2: Solve the system of equations.

We can use either substitution or elimination to solve this system. Let's use the elimination method.

#### Multiply the first equation by 2:
\[
2(50x + 90y) = 2(286)
\]
\[
100x + 180y = 572
\]

Now, subtract the second equation from this new equation:
\[
(100x + 180y) - (100x + 70y) = 572 - 418
\]
\[
110y = 154
\]
\[
y = \frac{154}{110} = 1.4
\]

### Step 3: Substitute $ y = 1.4 $ into one of the original equations.

Using the first equation:
\[
50x + 90(1.4) = 286
\]
\[
50x + 126 = 286
\]
\[
50x = 286 - 126 = 160
\]
\[
x = \frac{160}{50} = 3.2
\]

### Step 4: Conclusion

- The cost per foot of redwood is **$3.20**.
- The cost per foot of pine is **$1.40**.

Would you like more details or have any questions?

Here are 5 related questions to further explore:
1. How would you solve this using substitution instead of elimination?
2. If the prices for both types of wood increased by 10%, what would be the new prices per foot?
3. How would the solution change if 150 feet of redwood and 60 feet of pine were purchased instead?
4. Can you generalize a method for solving any system of two linear equations like this?
5. What would happen if the cost total in the second purchase changed to $500?

**Tip:** When solving systems of equations, elimination works well when the coefficients of one variable are already aligned for subtraction or easy to align through multiplication.

A carpenter purchased 50 ft of redwood and 90 ft of pine for a total cost of $286. A second purchase, at the same prices, included 100 ft of redwood and 70 ft of pine for a total cost of $418. Find the cost per foot of redwood and of pine.

Learn how to solve a system of linear equations to find the cost per foot of redwood and pine. This problem involves purchasing 50 ft of redwood and 90 ft of pine for $286, and 100 ft of redwood and 70 ft of pine for $418. Ideal for students in Grades 8-10.

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