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=3x
y equals 3 x

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

y=x

Answer 1

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

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

\[ m = \frac{y₂ - y₁}{x₂ - x₁} \]

Using the points (3, 3) (let's call this point A) and (8, 8) (let's call this point B):

• \(x₁ = 3\), \(y₁ = 3\)
• \(x₂ = 8\), \(y₂ = 8\)

Calculating the slope:

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

Now, since we want the equation of the line through the origin (0, 0) with the slope \(m\), we can use the slope-intercept form:

\[ y = mx \]

Substituting \(m = 1\):

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

Thus, the equation of the line through the origin is:

\[ y = x \]

Therefore, the correct response is:

y = x