A worker at a farm supply company in Franklin County is mixing up a 420-kilogram batch of its special grain blend for horses, which costs $5.80 per kilogram to make. If the mixture includes oats that cost $9.40 per kilogram and feed corn that costs $3.40 per kilogram, how many kilograms of each should the worker include in the mixture?

We can solve this problem by using a system of linear equations. Let's define the variables:

- Let $ x $ be the number of kilograms of oats (which cost $9.40 per kilogram).
- Let $ y $ be the number of kilograms of feed corn (which cost $3.40 per kilogram).

### Step 1: Set up the equations
The total weight of the mixture is 420 kilograms:
\[
x + y = 420
\]

The total cost of the mixture is $5.80 per kilogram. So, the total cost for 420 kilograms is:
\[
9.40x + 3.40y = 5.80 \times 420
\]
\[
9.40x + 3.40y = 2436
\]

### Step 2: Solve the system of equations

Now we have two equations:
1. $ x + y = 420 $
2. $ 9.40x + 3.40y = 2436 $

We can solve this system by substitution or elimination. Let's use substitution.

From the first equation, solve for $ y $:
\[
y = 420 - x
\]

Substitute this expression for $ y $ into the second equation:
\[
9.40x + 3.40(420 - x) = 2436
\]

### Step 3: Simplify and solve for $ x $

Distribute $ 3.40 $ on the left-hand side:
\[
9.40x + 3.40 \times 420 - 3.40x = 2436
\]
\[
9.40x + 1428 - 3.40x = 2436
\]
Combine like terms:
\[
6x + 1428 = 2436
\]
Subtract 1428 from both sides:
\[
6x = 1008
\]
Divide by 6:
\[
x = 168
\]

### Step 4: Solve for $ y $

Now that we know $ x = 168 $, substitute it back into the equation $ y = 420 - x $:
\[
y = 420 - 168 = 252
\]

### Final Answer:
- The worker should include **168 kilograms of oats** and **252 kilograms of feed corn** in the mixture.

Would you like more details on any step, or do you have any questions?

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Here are 5 related questions to deepen your understanding:

1. How would the proportions change if the total cost per kilogram of the blend were higher?
2. Can you set up the problem differently using elimination instead of substitution?
3. How does the price difference between oats and corn affect the final mixture proportions?
4. What if the mixture had to include an additional ingredient with a different cost per kilogram?
5. How would you check if the solution satisfies both the weight and cost constraints?

**Tip:** When solving a system of equations, always check that your solution satisfies both original equations to ensure correctness.

Learn how to solve a farm supply problem involving a 420-kilogram grain blend mixture, using oats and feed corn with different costs. This detailed step-by-step solution explores key algebraic concepts and the substitution method for solving systems of equations. Suitable for Grades 9-10.

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