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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)Responsesy=8xy equals 8 xy=3xy equals 3 xy=33xy equals Start Fraction 3 over 3 End Fraction xy=x

9 answers

To determine the equation of the line passing through the origin that would be in the form of \(y = mx\), we first need to find the slope \(m\) of the given line that passes through the points (3, 3) and (8, 8).

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

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

Substituting the points (3, 3) and (8, 8) into the formula:

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

Since the slope \(m\) is 1, the equation of the line passing through the origin is:

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

Thus, the answer 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=36x
y equals 36 x

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

y=12x
y equals 12 x

y=2x

To derive the equation of the line passing through the origin based on the graph described, we can identify the slope of the line, which represents the ratio of inches to feet.

From the points given on the line:

  • At \(x = 1\) foot, \(y = 12\) inches.
  • At \(x = 2\) feet, \(y = 24\) inches.
  • At \(x = 3\) feet, \(y = 36\) inches.

This shows a consistent pattern where for every 1 foot increase, the inches increase by 12. Therefore, the slope \(m\) can be calculated as follows:

\[ m = \frac{\text{Change in } y}{\text{Change in } x} = \frac{12 \text{ inches}}{1 \text{ foot}} = 12 \]

Since the relationship is directly proportional and passes through the origin, the equation of the line is:

\[ y = mx = 12x \]

Thus, the answer 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, 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), (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), (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)

To complete the ordered pairs in the table using the equation \(y = 5x\), we will plug in the values of \(x\) into the equation to find the corresponding \(y\) values.

Here's the calculation for each \(x\):

  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 table of ordered pairs is:

\[ (0, 0), (1, 5), (2, 10) \]

The correct response is:

\[ (0, 0), (1, 5), (2, 10) \]

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

To graph the proportional relationship \(y = -2x\), we will determine the points that satisfy this equation based on various \(x\) values.

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

With these calculations, the three points that we will plot on the graph would be:

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

Based on the response options you provided:

The correct plotted points corresponding to the equation \(y = -2x\) are:

\[ (0, 0), (1, -2), (2, -4) \]

So the response that matches this is:

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.

Use the image to answer the question.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 two plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis and left parenthesis 5 comma 1 right parenthesis. Is this the correct graph of the equation y=15x ?(1 point)ResponsesYes, it is the correct graph because the ordered pair (0, 0) satisfies the equation and the slope m=riserun=15.Yes, it is the correct graph because the ordered pair left parenthesis 0 comma 0 right parenthesis satisfies the equation and the slope m equals Start Fraction rise over run End Fraction equals Start Fraction 1 over 5 End Fraction .Yes, it is the correct graph because the slope =51.Yes, it is the correct graph because the slope equals Start Fraction 5 over 1 End Fraction .No, it is not the correct graph because the slope m=riserun=15.No, it is not the correct graph because the slope m equals Start Fraction rise over run End Fraction equals Start Fraction 1 over 5 End Fraction .No, it is not the correct graph because the ordered pair (1, 5) is not a point on the graph of y=15x

To determine if the graph is correct for the equation \(y = \frac{1}{5}x\) (not \(y = 15x\)), we analyze the two points plotted on the graph: \((0, 0)\) and \((5, 1)\).

  1. Checking if \((0, 0)\) satisfies the equation:

    • When \(x = 0\): \[ y = \frac{1}{5}(0) = 0 \]
    • Thus, \((0, 0)\) does indeed satisfy the equation.

  2. Checking if \((5, 1)\) satisfies the equation:

    • When \(x = 5\): \[ y = \frac{1}{5}(5) = 1 \]
    • Therefore, \((5, 1)\) also satisfies the equation.

  3. Calculating the slope:

    • The slope \(m\) is calculated as: \[ m = \frac{\text{rise}}{\text{run}} = \frac{1 - 0}{5 - 0} = \frac{1}{5} \]

Thus, the slope is \(\frac{1}{5}\), and this indicates the correct equation should be \(y = \frac{1}{5}x\).

Given the analysis above, we find that:

  • The correct response to the question is:

"No, it is not the correct graph because the slope \(m = \frac{\text{rise}}{\text{run}} = \frac{1}{5}\)."

This indicates the graph shown does not represent \(y = 15x\); instead, it represents \(y = \frac{1}{5}x\).