Question

use the image described 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.

Answer 1

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

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

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

Substituting the coordinates of the points into the formula:

- Let \( (x_1, y_1) = (3, 3) \) and \( (x_2, y_2) = (8, 8) \).

Now substitute these values into the slope formula:

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

The slope \( m \) is 1. Therefore, the equation of the line in slope-intercept form that goes through the origin (0,0) is:

\[
y = mx
\]

Substituting \( m = 1 \):

\[
y = 1x
\]

or simply:

\[
y = x
\]

Thus, the equation of the line through the origin is \( y = x \).