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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 answer

To derive the equation of the line \( y = mx \) that passes through the origin and is based on the given points, we first need to determine the slope \( m \) of the line using the two 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 in the coordinates of the points:

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

Now, substitute these values into the slope formula:

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

Now that we have the slope \( m = 1 \), we can substitute it into the equation of the line that passes through the origin:

\[ y = mx \implies y = 1x \implies y = x \]

Thus, the equation of the line is:

\[ \boxed{y = x} \]