Use the image to answer the question.

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=3x
y equals 3 x

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

y=x
y equals x

y=8x
y equals 8 x
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Answer 1

To derive the equation of the line using the points (3, 3) and (8, 8), we can start by finding the slope (m) of the line.

The formula for the slope between two points (x₁, y₁) and (x₂, y₂) is:

\[ m = \frac{y₂ - y₁}{x₂ - x₁} \]

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

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

Now that we have the slope (m = 1), we can express the equation of the line in the form of \( y = mx \):

\[ y = 1x \]

This simplifies to:

\[ y = x \]

Thus, the derived equation for the line that passes through the origin is:

y = x