Linear Systems Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve both examples from the worksheet step by step.

---

Example 1: Penelope’s School Supplies

Problem:
Penelope is buying school supplies. Spiral notebooks are $4 each and binder notebooks are $5 each. She needs at least 8 notebooks but wants to spend less than $40. Solve the system by graphing and list a possible solution.

---

#### Step 1: Define the two variables
Let:
- $ x $ = number of spiral notebooks
- $ y $ = number of binder notebooks

---

#### Step 2: Write the two inequalities

1. At least 8 notebooks:
This means total notebooks ≥ 8
$$
x + y \geq 8
$$

2. Spend less than $40:
Cost: $4x + 5y < 40$
$$
4x + 5y < 40
$$

So, the system is:
$$
\begin{cases}
x + y \geq 8 \\
4x + 5y < 40
\end{cases}
$$

---

#### Step 3: Graph the system

We'll graph both inequalities on the coordinate plane.

##### Graph 1: $ x + y \geq 8 $

- First, graph the line $ x + y = 8 $
- Find intercepts:
- When $ x = 0 $, $ y = 8 $
- When $ y = 0 $, $ x = 8 $
- Draw a solid line (because it's ≥)
- Shade above the line (since $ x + y \geq 8 $)

##### Graph 2: $ 4x + 5y < 40 $

- Graph the line $ 4x + 5y = 40 $
- Find intercepts:
- When $ x = 0 $: $ 5y = 40 $ → $ y = 8 $
- When $ y = 0 $: $ 4x = 40 $ → $ x = 10 $
- Draw a dashed line (because it's <)
- Shade below the line (since $ 4x + 5y < 40 $)

The solution region is where the shaded areas overlap.

---

#### Step 4: Find a possible solution

We need a point in the overlapping shaded region that satisfies both inequalities.

Try $ x = 5 $, $ y = 4 $:

- Check $ x + y = 5 + 4 = 9 \geq 8 $ ✔
- Check $ 4(5) + 5(4) = 20 + 20 = 40 $ → But we need less than $40 → 40 is not less than 40 ✘

Try $ x = 6 $, $ y = 3 $:

- $ x + y = 9 \geq 8 $ ✔
- $ 4(6) + 5(3) = 24 + 15 = 39 < 40 $ ✔

✔ Valid solution!

Another possibility: $ x = 4 $, $ y = 5 $

- $ 4 + 5 = 9 \geq 8 $ ✔
- $ 4(4) + 5(5) = 16 + 25 = 41 > 40 $ ✘

Try $ x = 7 $, $ y = 2 $

- $ 7 + 2 = 9 \geq 8 $ ✔
- $ 4(7) + 5(2) = 28 + 10 = 38 < 40 $ ✔

Also valid.

So one possible solution is: 6 spiral notebooks and 3 binder notebooks

---

✔ Answer for Example 1:

- Inequalities:
$$
x + y \geq 8 \\
4x + 5y < 40
$$
- Possible solution: $ (6, 3) $ → 6 spiral, 3 binder notebooks

---

Example 2: Restaurant Buying Vegetables

Problem:
A local restaurant is purchasing vegetables. Broccoli is $2.40/lb., celery is $3.30/lb. They want to buy at least 8 lbs of vegetables and spend at most $20. Can they buy 6 pounds of broccoli and 2 pounds of celery?

---

#### Step 1: Define variables
Let:
- $ x $ = pounds of broccoli
- $ y $ = pounds of celery

---

#### Step 2: Write inequalities

1. At least 8 lbs total:
$$
x + y \geq 8
$$

2. At most $20 spent:
$$
2.40x + 3.30y \leq 20
$$

So the system is:
$$
\begin{cases}
x + y \geq 8 \\
2.4x + 3.3y \leq 20
\end{cases}
$$

---

#### Step 3: Test the point (6, 2)

Check if $ (6, 2) $ satisfies both inequalities.

1. Total weight: $ 6 + 2 = 8 \geq 8 $ ✔

2. Cost: $ 2.40(6) + 3.30(2) = 14.40 + 6.60 = 21.00 $

But $ 21.00 > 20 $ → Exceeds budget ✘

So, no, they cannot buy 6 lbs of broccoli and 2 lbs of celery because it costs $21, which is more than $20.

---

✔ Answer for Example 2:

- No, they cannot buy 6 lbs of broccoli and 2 lbs of celery because the cost would be $21.00, which exceeds their $20 budget.

---

Summary of Steps (Fill in the blanks):

1. Define the two variables.
2. Write the two inequalities.
3. Solve by graphing. *Remember to label the x- and y-axis!*
4. Answer the question(s). *Make sure to read carefully!*

---

Final Answers:

Example 1:
- Inequalities: $ x + y \geq 8 $, $ 4x + 5y < 40 $
- Possible solution: 6 spiral, 3 binder notebooks

Example 2:
- No, they cannot buy 6 lbs of broccoli and 2 lbs of celery because it costs $21 > $20.

Let me know if you'd like the graphs sketched out!