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alt="An illustration shows a coordinate plane with 4 quadrants. The x-axis ranges from negative 9 to 9 in one unit increments, and the y-axis ranges from negative 11 to 11 in one unit increments. A line is graphed on the plane. An upward slanting line passes through points plotted at left parenthesis 1 comma 4 right parenthesis and left parenthesis 2 comma 8 right parenthesis."

Find the equation of the line y=mx
through the origin that describes the proportional relationship shown in the graph.

23 answers

To find the equation of the line \( y = mx \) that describes the proportional relationship, we first need to determine the slope \( m \) of the line that passes through the points \( (1, 4) \) and \( (2, 8) \).

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

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

Substituting the given points \( (1, 4) \) and \( (2, 8) \):

  • \( x_1 = 1 \), \( y_1 = 4 \)
  • \( x_2 = 2 \), \( y_2 = 8 \)

We can calculate the slope as follows:

\[ m = \frac{8 - 4}{2 - 1} = \frac{4}{1} = 4 \]

Now that we have determined the slope \( m \), the equation of the line that passes through the origin and describes the proportional relationship is:

\[ y = 4x \]

Thus, the equation of the line is:

\[ y = 4x \]

An illustration shows a graph labeled Football Cost. The horizontal axis is labeled Quantity and ranges from 0 to 5 in increments of 1. The vertical axis is labeled Cost and ranges from 0 to 39 in increments of 3. An upward slanting line segment connects points plotted at left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 9 right parenthesis, left parenthesis 2 comma 18 right parenthesis, left parenthesis 3 comma 27 right parenthesis, and left parenthesis 4 comma 36 right parenthesis.
The price for footballs is displayed in the graph. Derive the equation of the proportional relationship through the origin.

To derive the equation of the proportional relationship (a linear equation through the origin) based on the points provided on the graph, we can observe the relationship between the quantity of footballs and the cost.

The points given are:

  • \( (0, 0) \)
  • \( (1, 9) \)
  • \( (2, 18) \)
  • \( (3, 27) \)
  • \( (4, 36) \)

From these points, we can identify that the cost increases in consistent increments for each additional football purchased. To find the slope \( m \) of the line, we can use the formula:

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

Let’s take the points \( (1, 9) \) and \( (0, 0) \):

\[ m = \frac{9 - 0}{1 - 0} = \frac{9}{1} = 9 \]

We can confirm this slope by checking other points:

  • From \( (2, 18) \) to \( (1, 9) \):

\[ m = \frac{18 - 9}{2 - 1} = \frac{9}{1} = 9 \]

  • From \( (3, 27) \) to \( (2, 18) \):

\[ m = \frac{27 - 18}{3 - 2} = \frac{9}{1} = 9 \]

  • From \( (4, 36) \) to \( (3, 27) \):

\[ m = \frac{36 - 27}{4 - 3} = \frac{9}{1} = 9 \]

Since the slope is consistent at \( 9 \), this indicates a proportional relationship between the quantity and cost.

The equation of the line through the origin can be expressed as:

\[ y = mx \]

Substituting the slope \( m = 9 \):

\[ y = 9x \]

Therefore, the equation of the proportional relationship through the origin is:

\[ y = 9x \]

This equation indicates that the cost \( y \) of footballs is \( 9 \) times the quantity \( x \).

Use the table to answer the question.

x
y
(x, y)

0 (0, )
4 (4, )
8 (8, )
Complete the ordered pairs in the table by finding the y
-coordinate of the ordered pairs for the equation y=14x
.

(2 points)
x
y
(x, y)
0 (0,
)
4 (4,
)
8 (8,
)

To complete the table for the equation \( y = 14x \), we can substitute the \( x \) values provided in the table into the equation to find their corresponding \( y \) values.

  1. For \( x = 0 \): \[ y = 14 \times 0 = 0 \] So the ordered pair is \( (0, 0) \).

  2. For \( x = 4 \): \[ y = 14 \times 4 = 56 \] So the ordered pair is \( (4, 56) \).

  3. For \( x = 8 \): \[ y = 14 \times 8 = 112 \] So the ordered pair is \( (8, 112) \).

The completed table with the ordered pairs is:

\[ \begin{array}{|c|c|c|} \hline x & y & (x, y) \ \hline 0 & 0 & (0, 0) \ 4 & 56 & (4, 56) \ 8 & 112 & (8, 112) \ \hline \end{array} \]

So the resulting ordered pairs are \( (0, 0) \), \( (4, 56) \), and \( (8, 112) \).

actually its 1/4

Thank you for clarifying! If the equation is actually \( y = \frac{1}{4}x \), we can use that equation to find the \( y \)-coordinates for the given \( x \) values in the table.

  1. For \( x = 0 \): \[ y = \frac{1}{4} \times 0 = 0 \] So the ordered pair is \( (0, 0) \).

  2. For \( x = 4 \): \[ y = \frac{1}{4} \times 4 = 1 \] So the ordered pair is \( (4, 1) \).

  3. For \( x = 8 \): \[ y = \frac{1}{4} \times 8 = 2 \] So the ordered pair is \( (8, 2) \).

Now the completed table with the ordered pairs is:

\[ \begin{array}{|c|c|c|} \hline x & y & (x, y) \ \hline 0 & 0 & (0, 0) \ 4 & 1 & (4, 1) \ 8 & 2 & (8, 2) \ \hline \end{array} \]

The resulting ordered pairs are \( (0, 0) \), \( (4, 1) \), and \( (8, 2) \). Thank you for your patience!

In a cleaning mixture, the amount of water is directly proportional to the amount of ammonia. A mix with 5 gallons of water requires 10 teaspoons of ammonia. How many gallons of water are needed for 25 teaspoons of ammonia? Write your answer in decimal form.(1 point)

Given that the amount of water is directly proportional to the amount of ammonia, we can set up the relationship mathematically as follows:

\[ \text{Water} = k \times \text{Ammonia} \]

where \( k \) is the constant of proportionality.

From the problem, we know that 5 gallons of water requires 10 teaspoons of ammonia. Using these values, we can find \( k \):

\[ 5 = k \times 10 \]

Solving for \( k \):

\[ k = \frac{5}{10} = 0.5 \]

Now we have the relationship:

\[ \text{Water} = 0.5 \times \text{Ammonia} \]

Next, we need to find how many gallons of water are needed for 25 teaspoons of ammonia. Using our equation:

\[ \text{Water} = 0.5 \times 25 \]

Calculating this:

\[ \text{Water} = 12.5 \text{ gallons} \]

Thus, 12.5 gallons of water are needed for 25 teaspoons of ammonia.

An illustration shows a coordinate plane with 4 quadrants. The x-axis ranges from negative 9 to 9 in one unit increments, and the y-axis ranges from negative 11 to 11 in one unit increments. A line is graphed on the plane. An upward slanting line line passes through points plotted at left parenthesis 3 comma 3 right parenthesis and left parenthesis 8 comma 8 right parenthesis.Use the graph to derive the equation y=mx for a line through the origin. Simplify the numerical coefficient, if needed.

To derive the equation of the line \( y = mx \) that passes through the origin and is defined by the given points \( (3, 3) \) and \( (8, 8) \), we first need to determine the slope \( m \) of the line.

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

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

Using the points \( (3, 3) \) and \( (8, 8) \):

  • Let \( (x_1, y_1) = (3, 3) \)
  • Let \( (x_2, y_2) = (8, 8) \)

We can substitute these values into the slope formula:

\[ m = \frac{8 - 3}{8 - 3} = \frac{5}{5} = 1 \]

Now that we have determined the slope \( m \), the equation of the line that passes through the origin and can be expressed as:

\[ y = mx \]

Substituting \( m = 1 \):

\[ y = 1 \cdot x \] or simply: \[ y = x \]

Thus, the equation of the line that passes through the origin and is defined by the points \( (3, 3) \) and \( (8, 8) \) is:

\[ y = x \]

This indicates a direct proportional relationship between \( y \) and \( x \) where for every unit increase in \( x \), \( y \) also increases by the same amount.

An illustration shows a graph labeled Inches per Feet. The horizontal axis is labeled feet and ranges from 0 to 4 in increments of 1. The vertical axis is labeled Inches and ranges from 0 to 40 in increments of 4. An upward slanting line segment connects points plotted at left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 12 right parenthesis, left parenthesis 2 comma 24 right parenthesis, and left parenthesis 3 comma 36 right parenthesis.
The proportional relationship between inches and feet is displayed on the graph. Derive the equation of the line through the origin.

(1 point)

To derive the equation of the line that represents the proportional relationship between inches and feet from the points given on the graph, we need to determine the slope of the line. The points provided are:

  • \( (0, 0) \)
  • \( (1, 12) \)
  • \( (2, 24) \)
  • \( (3, 36) \)

Since the line passes through the origin \( (0, 0) \), we can calculate the slope \( m \) using any two points. Let's use the points \( (1, 12) \) and \( (0, 0) \):

Step 1: Calculate the slope \( m \)

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

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

Using the points \( (1, 12) \) and \( (0, 0) \):

  • \( (x_1, y_1) = (0, 0) \)
  • \( (x_2, y_2) = (1, 12) \)

Now, substituting into the formula:

\[ m = \frac{12 - 0}{1 - 0} = \frac{12}{1} = 12 \]

Step 2: Write the equation

Since it is a proportional relationship and the line passes through the origin, we can write the equation in the form:

\[ y = mx \]

Substituting the slope \( m = 12 \):

\[ y = 12x \]

Conclusion

Thus, the equation of the line through the origin that represents the proportional relationship between inches and feet is:

\[ y = 12x \]

This means that for every 1 foot, there are 12 inches.

x y (x, y)
0 (0, )
1 (1, )
2 (2, )
Complete the ordered pairs in the table by finding the y -value of the ordered pairs for the equation y=5x .

To complete the table using the equation \( y = 5x \), we will substitute the given \( x \) values into the equation to find the corresponding \( y \) values.

  1. For \( x = 0 \): \[ y = 5 \times 0 = 0 \] So, the ordered pair is \( (0, 0) \).

  2. For \( x = 1 \): \[ y = 5 \times 1 = 5 \] So, the ordered pair is \( (1, 5) \).

  3. For \( x = 2 \): \[ y = 5 \times 2 = 10 \] So, the ordered pair is \( (2, 10) \).

Now, we can fill out the table:

\[ \begin{array}{|c|c|c|} \hline x & y & (x, y) \ \hline 0 & 0 & (0, 0) \ 1 & 5 & (1, 5) \ 2 & 10 & (2, 10) \ \hline \end{array} \]

So the completed ordered pairs are \( (0, 0) \), \( (1, 5) \), and \( (2, 10) \).

Graph the proportional relationship y=−2x

To graph the proportional relationship given by the equation \( y = -2x \), follow these steps:

Step 1: Determine Key Points

You can find some key points by substituting different values of \( x \) into the equation \( y = -2x \).

  • When \( x = 0 \): \[ y = -2(0) = 0 \quad \Rightarrow \quad (0, 0) \]

  • When \( x = 1 \): \[ y = -2(1) = -2 \quad \Rightarrow \quad (1, -2) \]

  • When \( x = -1 \): \[ y = -2(-1) = 2 \quad \Rightarrow \quad (-1, 2) \]

  • When \( x = 2 \): \[ y = -2(2) = -4 \quad \Rightarrow \quad (2, -4) \]

  • When \( x = -2 \): \[ y = -2(-2) = 4 \quad \Rightarrow \quad (-2, 4) \]

Step 2: Plot the Points

Now, plot the points on the coordinate plane:

  • \( (0, 0) \) (the origin)
  • \( (1, -2) \)
  • \( (-1, 2) \)
  • \( (2, -4) \)
  • \( (-2, 4) \)

Step 3: Draw the Line

Once you plot these points, you will see a straight line passing through them since the relationship is linear. Because the slope \( m = -2 \), the line will slope downwards from left to right.

Characteristics of the Graph

  • The line crosses the origin \( (0, 0) \).
  • The negative slope indicates that for every increase of 1 unit in \( x \), \( y \) decreases by 2 units.
  • The graph extends infinitely in both directions.

Final Representation

If you were to sketch this on graph paper or a digital tool, your line would look like this:

|         *
|      *
|   *
|*  
|________________
    *    *     *

(Note: This is a simple textual representation. Ideally, you would make a graph on a coordinate plane.)

The line should extend infinitely with the pattern established by the calculated points. The equation \( y = -2x \) represents a linear decrease with a slope of -2, meaning it is a proportional relationship that is inverse in nature.

what would it look like labled

To graph the equation \( y = -2x \) with proper labels, the graph would have both axes labeled and a clear indication of the line itself. Here’s a textual representation of what the graph would roughly look like, along with labels for the axes and points.

       y
      ^          *
      |      *
  4   |  *               
      |*                    
  2   |                        
      |________________________> x
     -2    -1    0    1    2  

-4   |
      |        *
      |
      |

Components of the Graph:

  • Axes:

    • The vertical axis labeled y, extending from -5 to 5 (or more, depending on your range).
    • The horizontal axis labeled x, extending from -3 to 3 (or more).

  • Key Points:

    • Points plotted at \( (0, 0) \), \( (1, -2) \), \( (-1, 2) \), \( (2, -4) \), \( (-2, 4) \) should be marked clearly.

  • Line:

    • A straight line connecting all points, showing that \( y \) decreases as \( x \) increases.

  • Labels:

    • On the graph, you could label the equation of the line:
      • Place a label near the line: "y = -2x".

Labeled Graph Example:

If you were to create this graph using a tool or on graph paper, it would look more organized with properly spaced axes, tick marks, and ideally, labeled coordinates as well.

For example, near the points, you could write:

  • At \( (0, 0) \): (0, 0)
  • At \( (1, -2) \): (1, -2)
  • At \( (-1, 2) \): (-1, 2)
  • At \( (2, -4) \): (2, -4)
  • At \( (-2, 4) \): (-2, 4)

Using Graphing Software:

If you use any graphing software or tools like Desmos or GeoGebra, setting up the equation \( y = -2x \) will automatically format and label the graph appropriately for clarity, making it easier to read and interpret.

no in a format like this: (7,9)

Sure! Here’s a more structured representation of the graph with specific points formatted in the style you requested:

Labeled Points on the Graph of \(y = -2x\)

| Point | Coordinates | |-------------|-------------| | Point A | (0, 0) | | Point B | (1, -2) | | Point C | (-1, 2) | | Point D | (2, -4) | | Point E | (-2, 4) |

Graph Visualization:

Here’s a simple representation of the line along with the labeled points:

      y
      ^
      |
  4   |           E (-2, 4)
      |       *  
  2   |  C (-1, 2)
      |  *
  0   |---------------------> x
      |  A (0, 0)
      |       B (1, -2)
 -2   |       *      
      |            D (2, -4)
 -4   |

Description:

  • A (0, 0): The line crosses the origin.
  • B (1, -2): When \( x = 1 \), \( y \) decreases to -2.
  • C (-1, 2): When \( x = -1 \), \( y \) increases to 2.
  • D (2, -4): When \( x = 2 \), \( y \) further decreases to -4.
  • E (-2, 4): When \( x = -2 \), \( y \) increases to 4.

You can plot these points on a coordinate plane, and draw a straight line through them to represent the equation \( y = -2x \).

A caterer has determined that the amount of meat to order for a particular entrée varies proportionally to the number of guests. If there are 25 guests, the caterer needs to order 40 pounds of meat. How much meat should the caterer order if there are 100 guests?(1 point)

To find out how much meat the caterer should order for 100 guests based on the proportional relationship, we can set up a proportion using the information given.

We know that:

  • For 25 guests, the caterer orders 40 pounds of meat.

Let \( x \) represent the amount of meat needed for 100 guests. We set up the following proportion:

\[ \frac{40 \text{ pounds}}{25 \text{ guests}} = \frac{x \text{ pounds}}{100 \text{ guests}} \]

Now we can cross-multiply to solve for \( x \):

\[ 40 \times 100 = 25 \times x \]

\[ 4000 = 25x \]

Now, divide both sides by 25 to find \( x \):

\[ x = \frac{4000}{25} = 160 \]

Thus, the caterer should order 160 pounds of meat for 100 guests.