An illustration shows Quadrant 1 of a coordinate plane. The x-axis is labeled Ounces and ranges from 0 to 15 in 3 unit increments. The y-axis ranges from 0 to 153 in 9 unit increments. The graph is titled Calories per Ounce. A line connects five points plotted on the graph. The coordinates of the plotted points are left parenthesis 0 comma 0 right parenthesis, left parenthesis 3 comma 36 right parenthesis, left parenthesis 6 comma 72 right parenthesis, left parenthesis 9 comma 108 right parenthesis, and left parenthesis 12 comma 144 right parenthesis.

Question

The proportional relationship between calories and ounces of soda is shown in the graph. How many calories are in 1 ounce?

Answer 1

To find the number of calories in 1 ounce of soda, we can analyze the coordinates of the points given in the graph. We know from the points provided that there is a consistent proportional relationship between ounces and calories.

The points plotted are:

• (0, 0)
• (3, 36)
• (6, 72)
• (9, 108)
• (12, 144)

From this, we can observe that as the ounces (x-coordinate) increase, the calories (y-coordinate) increase proportionately.

To find the calories per ounce, we can look at the values. For example, at 3 ounces, there are 36 calories. To find the calories per ounce, we divide the calories by the ounces:

\[ \frac{36 \text{ calories}}{3 \text{ ounces}} = 12 \text{ calories per ounce} \]

To verify this, we can check another point on the line, like from the (6, 72) point:

\[ \frac{72 \text{ calories}}{6 \text{ ounces}} = 12 \text{ calories per ounce} \]

And one more, at (9, 108):

\[ \frac{108 \text{ calories}}{9 \text{ ounces}} = 12 \text{ calories per ounce} \]

Since all calculations yield the same result, we confirm that there are:

12 calories in 1 ounce of soda.

Answer 2

alt="An illustration shows a coordinate plane with the x-axis ranging from negative 9 to 9 in one unit increments, and the y-axis ranging from negative 11 to 11 in one unit increments. A solid line passes through four plotted points and extends beyond. A solid line with arrows at both ends passes through points with coordinates left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 3 right parenthesis, left parenthesis 2 comma 6 right parenthesis, and left parenthesis 3 comma 9 right parenthesis. The solid line forms the hypotenuse for two triangles. The first triangle is marked with a dotted line and has vertices with coordinates left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 0 right parenthesis, and left parenthesis 1 comma 3 right parenthesis. The base is labeled as 1, and the perpendicular height on the right is labeled as 3. The second triangle is marked with a dashed line and has vertices with coordinates left parenthesis 1 comma 3 right parenthesis, left parenthesis 3 comma 3 right parenthesis, and left parenthesis 3 comma 9 right parenthesis. The base is labeled as 2, and the perpendicular height on the right is labeled as 6."
Use similar triangles to determine the slope of the line.

Answer 3

To determine the slope of the line based on the plotted points and the triangles described, we can use the coordinates of two points on the line. The line connects the points (0, 0), (1, 3), (2, 6), and (3, 9).

The slope \( m \) of a line is calculated using the formula:

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

For example, we can use the points (1, 3) and (0, 0):

• Let \( (x_1, y_1) = (0, 0) \)
• Let \( (x_2, y_2) = (1, 3) \)

Substituting the values into the slope formula:

\[ m = \frac{3 - 0}{1 - 0} = \frac{3}{1} = 3 \]

This means the slope between those two points is 3.

We can check this slope using another pair of points, for instance (2, 6) and (1, 3):

• Let \( (x_1, y_1) = (1, 3) \)
• Let \( (x_2, y_2) = (2, 6) \)

Now calculating the slope:

\[ m = \frac{6 - 3}{2 - 1} = \frac{3}{1} = 3 \]

Lastly, checking the slope between (3, 9) and (2, 6):

• Let \( (x_1, y_1) = (2, 6) \)
• Let \( (x_2, y_2) = (3, 9) \)

Calculating the slope again:

\[ m = \frac{9 - 6}{3 - 2} = \frac{3}{1} = 3 \]

Since we got the same slope of 3 from different pairs of points, we can conclude confidently that the slope of the line is:

3.

Answer 4

Use the table to answer the question.

($) Cost of Candy Bar A
Quantity ($) Cost
1 1.25
2 2.50
3 3.75
4 5.00
5 6.25
The cost of Candy Bar A is displayed in the table. The proportional relationship between the cost and quantity for Candy Bar B is represented by the equation y=x
. Which candy bar is the cheapest cost per bar? Enter 1 for Candy Bar A. Enter 2 for Candy Bar B.

Answer 5

To determine which candy bar has the cheapest cost per bar, we need to calculate the cost per bar for both Candy Bar A (as seen in the table) and Candy Bar B (given by the equation \( y = x \), indicating that the cost is equal to the quantity).

Candy Bar A

From the table, we can calculate the cost per bar as follows:

• For 1 candy bar: Cost = $1.25 → Cost per bar = \( \frac{1.25}{1} = 1.25 \)
• For 2 candy bars: Cost = $2.50 → Cost per bar = \( \frac{2.50}{2} = 1.25 \)
• For 3 candy bars: Cost = $3.75 → Cost per bar = \( \frac{3.75}{3} = 1.25 \)
• For 4 candy bars: Cost = $5.00 → Cost per bar = \( \frac{5.00}{4} = 1.25 \)
• For 5 candy bars: Cost = $6.25 → Cost per bar = \( \frac{6.25}{5} = 1.25 \)

So for Candy Bar A, the cost per bar is consistently $1.25.

Candy Bar B

The equation \( y = x \) implies that the cost per bar is $1.00, regardless of the quantity purchased.

Let's confirm:

• For 1 candy bar: Cost = $1 → Cost per bar = \( \frac{1}{1} = 1.00 \)
• For 2 candy bars: Cost = $2 → Cost per bar = \( \frac{2}{2} = 1.00 \)
• For 3 candy bars: Cost = $3 → Cost per bar = \( \frac{3}{3} = 1.00 \)

So for Candy Bar B, the cost per bar is consistently $1.00.

Conclusion

Comparing the two:

• Candy Bar A: Cost per bar = $1.25
• Candy Bar B: Cost per bar = $1.00

The cheaper option is Candy Bar B.

Answer: 2 (for Candy Bar B)

Answer 6

An illustration of a coordinate plane with four quadrants shows the x-axis ranging from negative 9 to 9 and the y-axis ranging from negative 11 to 11 in increments of 1. A solid upward line with arrows at both ends is plotted on the graph. The line passes through the plotted points left parenthesis 0 comma negative 2 right parenthesis and left parenthesis 1 comma 3 right parenthesis.Use the graph to derive the equation of the line in slope-intercept form.

Answer 7

To derive the equation of the line in slope-intercept form, which is given by the formula \( y = mx + b \) (where \( m \) is the slope and \( b \) is the y-intercept), we first need to calculate the slope of the line using the two points:

1. Identify the points:

   • Point 1: \((0, -2)\)
   • Point 2: \((1, 3)\)

2. Calculate the slope (\( m \)): The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

   Applying the coordinates of the given points:

   \[ m = \frac{3 - (-2)}{1 - 0} = \frac{3 + 2}{1 - 0} = \frac{5}{1} = 5 \]

3. Use the slope and a point to find the y-intercept (\( b \)): Now that we have the slope \( m = 5 \), we can use one of the points to solve for \( b \). Let’s use the point \((0, -2)\):

   \[ y = mx + b \] Substitute \( y = -2 \), \( m = 5 \), and \( x = 0 \):

   \[ -2 = 5(0) + b \]

   This simplifies to:

   \[ -2 = 0 + b \implies b = -2 \]

4. Form the equation: Now that we have both \( m \) and \( b \), we can write the equation of the line:

   \[ y = 5x - 2 \]

Conclusion

The equation of the line in slope-intercept form is:

\[ \boxed{y = 5x - 2} \]

Answer 8

Use the table to answer the question.

x
y

−5 0
0 5
2 7
Which linear equation represents this nonproportional linear relationship?

(1 point)
Responses

y=x+5
y equals x plus 5

y=−x+5
y equals negative x plus 5

y=1/5x+5
y equals Start Fraction 1 over 5 End Fraction x plus 5

y=−5x+5

Answer 9

To find the linear equation that represents the nonproportional linear relationship shown in the table, we will determine the slope using the given points and the y-intercept.

Based on the table, we have the following points:

1. \((-5, 0)\)
2. \((0, 5)\)
3. \((2, 7)\)

Step 1: Calculate the slope (m)

Using the points (0, 5) and (2, 7):

Let \((x_1, y_1) = (0, 5)\) and \((x_2, y_2) = (2, 7)\).

The slope \( m \) is calculated as follows:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 5}{2 - 0} = \frac{2}{2} = 1 \]

Step 2: Write the equation in point-slope form

With the slope \( m = 1 \) and using the y-intercept found at (0, 5) (where \( y = 5 \) when \( x = 0 \)), the equation of the line in slope-intercept form \( y = mx + b \) can be written as:

\[ y = 1x + 5 \quad \text{or simply} \quad y = x + 5 \]

Conclusion

The correct equation representing the nonproportional linear relationship from the table is:

\[ \boxed{y = x + 5} \]

Answer 10

The table shows the number of parts, y
, a machine produces each hour, x
. Write an equation for this situation, and indicate if this is proportional or nonproportional.

y
(number of parts) x
(number of hours)
0 0
125 1
250 2
375 3
500 4
(1 point)
Responses

y=x+125
, nonproportional
y equals x plus 125 , nonproportional

y=125x+1
, nonproportional
y equals 125 x plus 1 , nonproportional

y=125x
, proportional
y equals 125 x , proportional

y=125x
, nonproportional

Answer 11

To analyze the situation presented in the table and derive the appropriate equation, let's first summarize the points:

• (0, 0): 0 parts in 0 hours
• (1, 125): 125 parts in 1 hour
• (2, 250): 250 parts in 2 hours
• (3, 375): 375 parts in 3 hours
• (4, 500): 500 parts in 4 hours

Step 1: Identify the relationship

We can see that as \( x \) (the number of hours) increases, \( y \) (the number of parts produced) increases consistently. The number of parts produced each hour is constant at 125 parts per hour.

Step 2: Derive the equation

The relationship can be expressed with the equation:

\[ y = 125x \]

This captures that for every hour \( x \), 125 parts are produced.

Step 3: Determine if it is proportional or nonproportional

A relationship is proportional if it passes through the origin (0, 0) and can be expressed in the form \( y = kx \), where \( k \) is a constant. Since the equation we derived is \( y = 125x \), and it passes through the origin, this means it is proportional.

Conclusion

The correct equation for this situation is:

\[ \boxed{y = 125x, \text{ proportional}} \]

Answer 12

Write y−6x=5
in slope-intercept form(1 point)

Answer 13

To convert the equation \( y - 6x = 5 \) into slope-intercept form, we need to solve for \( y \) in terms of \( x \). The slope-intercept form is given by the formula:

\[ y = mx + b \]

where \( m \) is the slope and \( b \) is the y-intercept.

Step 1: Isolate \( y \)

Starting with the given equation:

\[ y - 6x = 5 \]

Add \( 6x \) to both sides to isolate \( y \):

\[ y = 6x + 5 \]

Conclusion

Now the equation is in slope-intercept form. Therefore, the answer is:

\[ \boxed{y = 6x + 5} \]

Answer 14

Graph the equation x−y=6
using the form y=mx+b
.(1 point)

Answer 15

To graph the equation \( x - y = 6 \) in the slope-intercept form \( y = mx + b \), we first need to rearrange the equation into that form.

Step 1: Rearranging the equation

Starting with the original equation:

\[ x - y = 6 \]

To isolate \( y \), subtract \( x \) from both sides:

\[ -y = -x + 6 \]

Now, multiply each term by -1 to solve for \( y \):

\[ y = x - 6 \]

Step 2: Identify the slope and y-intercept

In the equation \( y = x - 6 \):

• The slope \( m = 1 \)
• The y-intercept \( b = -6 \)

Step 3: Plot the graph

1. Start with the y-intercept: Plot the point (0, -6) on the graph.

2. Use the slope to find another point: Since the slope is 1, for every increase of 1 in \( x \), \( y \) increases by 1.

   • From (0, -6):
     • When \( x = 1 \), \( y = 1 - 6 = -5 \) → Plot the point (1, -5).
     • When \( x = 2 \), \( y = 2 - 6 = -4 \) → Plot the point (2, -4).

3. Draw the line: Connect these points with a straight line, extending in both directions.

Conclusion

The graph of the equation \( x - y = 6 \) (or \( y = x - 6 \)) is a straight line that crosses the y-axis at (0, -6) and has a slope of 1.

You can represent the line graphically with the plotted points and draw the line between them. However, as a text-based program, I can't draw directly, but now you have the steps to do this on graph paper or a graphing tool!