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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 (3, 3) and (8, 8).

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=8x
y equals 8 x

y=3x
y equals 3 x

y=x
y equals x

y=33x

5 answers

To derive the equation of the line in the form \(y = mx\) that passes through the origin, we need to determine the slope \(m\) using the two points given: (3, 3) and (8, 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 in our points \((x_1, y_1) = (3, 3)\) and \((x_2, y_2) = (8, 8)\):

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

Thus, the slope \(m\) is 1. Therefore, the equation of the line through the origin is:

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

The correct response is:

y=x

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=2x
y equals 2 x

y=12x

To derive the equation of the line through the origin based on the points provided in the graph, we need to find the slope \(m\) using the given points. The points you provided are:

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

We can calculate the slope \(m\) using any two of these points. Let's use the points (1, 12) and (0, 0):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12 - 0}{1 - 0} = \frac{12}{1} = 12 \]

Now that we have the slope \(m\), we can write the equation of the line in the form \(y = mx\):

\[ y = 12x \]

The correct response is:

y=12x

Table, 3 catagories, x,y, and (x, y).

X contains

0

1

2

y is empty.

(x, y) contains

(0, ) corosponding to 0

(1, ) corresponding to 1

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

To find the \(y\)-values for the equation \(y = 5x\), we can substitute the values of \(x\) provided in the table.

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

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

The correct response is:

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