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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)
Responses

y=8x
y equals 8 x

y=3x
y equals 3 x

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

y=x
y equals x
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Answer 1

To find the equation of the line that passes through the origin and has the same slope as the line that passes through the points (3, 3) and (8, 8), we first need to determine the slope (m) of the line.

Using the formula for slope:

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

For the points (3, 3) and (8, 8):

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

Now plugging in the values:

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

This means the slope of the line is 1.

Now, since we want the equation of the line that passes through the origin (0, 0) using the slope-intercept form \(y = mx\), we substitute \(m\) into the equation:

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

So the final answer is:

y = x