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Rajan is filling a basket with pears of equal weight. Draw a graph and write an equation to represent this situation.

The table shows the relationship between the number of pears and the total weight of the basket. After 1 pear, the total weight of the basket is 12 ounces. After 4 pears, the total weight is 39 ounces. Plot the two points on the graph to show the total weight of the basket of pears after 1 pear and 4 pears are put in the basket.
Number of pears Total weight (oz)
1 12
4 39
You got it!
For each pear added to the basket, the total weight increases at a constant rate. So, we can draw a line through these two points to show the linear relationship between the number of pears and the total weight in ounces. 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
(1, 12)(4, 39)
0
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9
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39
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Number of pears
Total weight (oz)
y= mx + b

Slope =
Nice!
What does the slope of 9 represent in this situation?
x
y
(1, 12)(4, 39)
0
1
2
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5
0
3
6
9
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39
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Number of pears
Total weight (oz)
y= mx + b
Slope = 9

The weight of the basket increases 9 ounces for each pear.

The weight of the basket increases 1 ounce for every 9 pears.

The initial weight of the basket is 9 ounces.
Good work!
To represent the relationship between the number of pears and the total weight of the basket, a linear equation can be written in the form y=mx+b. In this equation, m is the slope of the line, or the rate of change of the weight of the basket. We just solved for and interpreted the slope. Substitute the slope for m in our equation.
x
y
(1, 12)(4, 39)
0
1
2
3
4
5
0
3
6
9
12
15
18
21
24
27
30
33
36
39
42
45
48
Number of pears
Total weight (oz)
y= mx + b
y=
x+b

Slope = 9
Nice!
Now let’s solve for b in our equation, or the vertical intercept. To solve for b, we can substitute the x and y-values for a point on the line into the equation. x represents the number of pears and y represents the total weight of the basket. Let’s use the point (1, 12). Start by substituting the values of x and y into the equation.
x
y
(1, 12)(4, 39)
0
1
2
3
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5
0
3
6
9
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30
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39
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48
Number of pears
Total weight (oz)
y
=
9x
+
b

12
=
9
·

1
+
b
You got it!
Now, solve for the value of b, or the vertical intercept.
x
y
(1, 12)(4, 39)
0
1
2
3
4
5
0
3
6
9
12
15
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27
30
33
36
39
42
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48
Number of pears
Total weight (oz)
y=9x+b

12
=
9·1+b

b
=
Good work!
What does b, or the vertical intercept, 3 represent in this situation?
x
y
(1, 12)(4, 39)
0
1
2
3
4
5
0
3
6
9
12
15
18
21
24
27
30
33
36
39
42
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48
Number of pears
Total weight (oz)
y
=
9x + b
b
=
3

The initial weight of the basket is 3 ounces.

There are 3 pears in the basket to begin with.

The weight of the basket increases 3 ounces for every 1 pear.

1 answer

To summarize the situation clearly and determine the linear equation, we start by establishing the relationship between the number of pears and their total weight in ounces:

Given Data Points:

  • After 1 pear, total weight = 12 ounces (Point 1: (1, 12))
  • After 4 pears, total weight = 39 ounces (Point 2: (4, 39))

Slope Calculation:

The slope (m) of the linear equation can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Where:

  • \( (x_1, y_1) = (1, 12) \)
  • \( (x_2, y_2) = (4, 39) \)

Calculating the slope: \[ m = \frac{39 - 12}{4 - 1} = \frac{27}{3} = 9 \]

Equation Format:

We now have \( m = 9 \). The linear equation has the form: \[ y = mx + b \] So far, we have: \[ y = 9x + b \]

Finding the Y-Intercept (b):

Using the point \( (1, 12) \) to find \( b \): \[ 12 = 9 \cdot 1 + b \] \[ 12 = 9 + b \] \[ b = 12 - 9 \] \[ b = 3 \]

Final Equation:

Now substituting \( b \) into the equation, we get: \[ y = 9x + 3 \]

Interpretation of Slope and Y-Intercept:

  • The slope (9) indicates that the total weight of the basket increases by 9 ounces for each additional pear added.
  • The y-intercept (3) represents the initial weight of the basket when there are no pears, meaning the basket itself weighs 3 ounces.

Conclusion:

Thus, the relationship between the number of pears and the total weight of the basket can be represented graphically with the plotted points and the equation: \[ y = 9x + 3 \]

Graphically, you can plot these points:

  • Point (1, 12) and Point (4, 39), then draw a line through them, illustrating that as the number of pears increases, the total weight in ounces increases linearly.