Connecting Equations to Graphs (Part 2)
Let's analyze different forms of linear equations and how the forms relate to
their graphs.
Bell Work:
Rewrite These!
Rewrite each quotient as a sum or a difference.
1.
5(x+10)
2.
11.2: Graphs of Two Equations
Here are two graphs that represent situations you have seen in earlier activities.
amount in tank (gallons)
500
450
400
うし
300
250
200
150
100
50
2 4
6
8 10 12 14 16 18 20 22 24 26
time (minutes)
1. The first graph represents a = 450 - 20t, which describes the relationship between
gallons of water in a tank and time in minutes.

Let's work through the problems step by step:

### Bell Work: Rewrite These Quotients

1. \( 5(x + 10) \)

To rewrite the expression as a sum or difference, distribute the 5 across the terms inside the parentheses:

\[
5(x + 10) = 5 \cdot x + 5 \cdot 10 = 5x + 50
\]

Thus, the quotient \( 5(x + 10) \) becomes the sum \( 5x + 50 \).

---

### 11.2: Graphs of Two Equations

For this part, we're analyzing the first equation and its corresponding graph. The given equation is:

\[
a = 450 - 20t
\]

Where:
- \( a \) is the amount of water in the tank (in gallons),
- \( t \) is the time (in minutes).

#### Analyzing the Equation:
- **Slope (-20):** The coefficient of \( t \) is -20, which means that the water in the tank is decreasing by 20 gallons per minute.
- **Y-intercept (450):** When \( t = 0 \), the amount of water in the tank is 450 gallons. This is the starting amount of water at time zero.

#### Graph Characteristics:
- **Y-intercept (450 gallons):** The graph starts at 450 gallons on the vertical axis (at time \( t = 0 \)).
- **Negative slope (-20):** The graph will have a negative slope, slanting downwards as time increases, representing the steady decrease of water in the tank over time.
- **X-intercept:** To find when the tank will be empty (when \( a = 0 \)), set the equation equal to zero and solve for \( t \):
  
\[
0 = 450 - 20t
\]

\[
20t = 450
\]

\[
t = \frac{450}{20} = 22.5
\]

So, the tank will be empty at \( t = 22.5 \) minutes. The x-intercept is at 22.5 minutes.

---

### Visualizing the Graph:
- The graph starts at 450 gallons (the y-intercept).
- It decreases linearly with a slope of -20 gallons per minute.
- The graph crosses the x-axis at \( t = 22.5 \), indicating when the tank is empty.

Would you like further details on how to interpret the second graph, or any clarification on how the equation relates to the graph?

### Five Related Questions:
1. What happens to the graph if the slope is steeper (e.g., -30 instead of -20)?
2. How would the graph change if the initial amount of water was greater, say 500 gallons?
3. Can you find the equation of a line parallel to \( a = 450 - 20t \)?
4. How would you solve for the time when the tank has 200 gallons left?
5. If water was added to the tank at 15 gallons per minute, how would the equation change?

### Tip:
The slope of a line in an equation tells you how quickly the quantity changes over time. A steeper negative slope means a faster decrease.

Connecting Equations to Graphs (Part 2)
Let's analyze different forms of linear equations and how the forms relate to their graphs.
Bell Work:
Rewrite These!
Rewrite each quotient as a sum or a difference.
1. 5(x+10)
2. 11.2: Graphs of Two Equations
Here are two graphs that represent situations you have seen in earlier activities.
amount in tank (gallons)
500
450
400
300
250
200
150
100
50
2 4
6
8 10 12 14 16 18 20 22 24 26
time (minutes)
1. The first graph represents a = 450 - 20t, which describes the relationship between gallons of water in a tank and time in minutes.

Analyze linear equations and how they relate to their graphs in this interactive lesson. Solve problems involving water levels in a tank and graph the equation a = 450 - 20t. Learn about slope, y-intercepts, and more. Perfect for students in Grades 8-10.

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