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=8x
y equals 8 x

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

y=3x
y equals 3 x

y=x
y equals x
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Answer 1

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

The slope \( m \) is calculated using the formula:

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

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

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

Now, since the line goes through the origin (0,0), the y-intercept is 0. Therefore, the equation of the line can be expressed as:

\[ y = mx \]

Substituting the value of \( m = 1 \):

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

Thus, the correct response is:

\[ y = x \]