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Use the image to answer the question.

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.

(1 point)
Responses

y=33x
y equals Start Fraction 3 over 3 End Fraction x

y=3x
y equals 3 x

y=x
y equals x

y=8x

7 answers

To derive the equation of the line passing through the origin (0,0) and having the same slope as the line that passes through the points (3, 3) and (8, 8), we first need to find the slope (m) of the line.

The slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

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

Substituting in the points \((3, 3)\) and \((8, 8)\):

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

So the slope \(m\) is 1.

Now, the equation of a line through the origin (0,0) with slope \(m\) is given by:

\[ y = mx \]

Substituting the value of \(m\):

\[ y = 1x \]

This simplifies to:

\[ y = x \]

Thus, the correct response is:

y = x

Use the image to answer the question.

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)
Responses

y=112x
y equals Start Fraction 1 over 12 End Fraction x

y=12x
y equals 12 x

y=2x
y equals 2 x

y=36x
y equals 36 x
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To derive the equation of the line that passes through the origin (0,0) based on the information provided, we first need to determine the slope (m) of the line. The given points on the line are (0, 0), (1, 12), (2, 24), and (3, 36).

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

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

We can take the points (0, 0) and (1, 12) to calculate the slope:

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

Thus, the slope \(m\) is 12.

The equation of a line through the origin (0,0) with slope \(m\) is expressed as:

\[ y = mx \]

Substituting the value of \(m\):

\[ y = 12x \]

Therefore, the correct response is:

y = 12x

Use the table to answer the question.

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 .

(1 point)
Responses

(0, 0), (1, 5), (3, 15)
left parenthesis 0 comma 0 right parenthesis , left parenthesis 1 comma 5 right parenthesis , left parenthesis 3 comma 15 right parenthesis

(0, 0), (1, 15) , (2, 25)
left parenthesis 0 comma 0 right parenthesis , left parenthesis 1 comma Start Fraction 1 over 5 End Fraction right parenthesis , left parenthesis 2 comma Start Fraction 2 over 5 End Fraction right parenthesis

(0, 0), (5, 1), (10, 2)
left parenthesis 0 comma 0 right parenthesis , left parenthesis 5 comma 1 right parenthesis , left parenthesis 10 comma 2 right parenthesis

(0, 0), (1, 5), (2, 10)
left parenthesis 0 comma 0 right parenthesis , left parenthesis 1 comma 5 right parenthesis , left parenthesis 2 comma 10 right parenthesis

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

  1. For \(x = 0\): \[ y = 5(0) = 0 \quad \Rightarrow \quad (0, 0) \]

  2. For \(x = 1\): \[ y = 5(1) = 5 \quad \Rightarrow \quad (1, 5) \]

  3. For \(x = 2\): \[ y = 5(2) = 10 \quad \Rightarrow \quad (2, 10) \]

Thus, the completed ordered pairs using the equation \(y = 5x\) are:

  • \((0, 0)\)
  • \((1, 5)\)
  • \((2, 10)\)

Therefore, the response is:

(0, 0), (1, 5), (2, 10)

Graph the proportional relationship y=−2x by plotting points.(1 point)
Responses

A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 5 right parenthesis, and left parenthesis 2 comma 10 right parenthesis.
Image with alt text: A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 5 right parenthesis, and left parenthesis 2 comma 10 right parenthesis.

A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 2 right parenthesis, and left parenthesis 2 comma 4 right parenthesis.
Image with alt text: A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 2 right parenthesis, and left parenthesis 2 comma 4 right parenthesis.

A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma negative 1 right parenthesis, and left parenthesis 2 comma negative 2 right parenthesis.
Image with alt text: A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma negative 1 right parenthesis, and left parenthesis 2 comma negative 2 right parenthesis.

A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma negative 2 right parenthesis, and left parenthesis 2 comma negative 4 right parenthesis.
Image with alt text: A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma negative 2 right parenthesis, and left parenthesis 2 comma negative 4 right parenthesis.
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To graph the proportional relationship given by the equation \(y = -2x\), you can find the corresponding \(y\) values for various \(x\) values.

Here are a few points calculated from the equation:

  1. For \(x = 0\): \[ y = -2(0) = 0 \quad \Rightarrow \quad (0, 0) \]

  2. For \(x = 1\): \[ y = -2(1) = -2 \quad \Rightarrow \quad (1, -2) \]

  3. For \(x = 2\): \[ y = -2(2) = -4 \quad \Rightarrow \quad (2, -4) \]

These points are:

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

Based on these calculations, the correct response for the ordered pairs plotted on the coordinate plane would be:

A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: (0, 0), (1, -2), and (2, -4).