Question

For each minute, the amount of water in the tank increases at a constant rate. So, we can draw a line through these two points to show the linear relationship between the number of minutes and the amount of water in the tank. Let’s write an equation to represent this line in the form y=mx+b. Start by finding m, or the slope. What is the slope of this line?
x
y
(3, 100)(7, 120)
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
100
120
140
160
180
200
Time (minutes)
Amount of Water (liters)
y= mx + b

Slope =

Answer 1

To find the slope \( m \) of the line that represents the relationship between time (minutes) and the amount of water (liters), we can use the formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\):

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Given the points:
- Point 1: \( (3, 100) \) \, (where \( x_1 = 3 \) and \( y_1 = 100 \))
- Point 2: \( (7, 120) \) \, (where \( x_2 = 7 \) and \( y_2 = 120 \))

Now we can substitute the values into the slope formula:

\[
m = \frac{120 - 100}{7 - 3}
\]
\[
m = \frac{20}{4} = 5
\]

So, the slope \( m \) is **5**.

This means that for every minute, the amount of water in the tank increases by 5 liters.

Now, if we want to write the equation in the form \( y = mx + b \), we also need to find the y-intercept \( b \). To find \( b \), we can use one of the points and substitute \( m \) back into the equation.

Using point \( (3, 100) \):

\[
y = mx + b \implies 100 = 5(3) + b
\]
\[
100 = 15 + b
\]
\[
b = 100 - 15 = 85
\]

Thus, the equation that represents the line is:

\[
y = 5x + 85
\]

In summary:
- Slope \( m = 5 \)
- Equation of the line: \( y = 5x + 85 \)