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Which equation represents the graph:

A graph of a linear function, f(x), is shown. The x-axis ranges from -3 to 7, and the y-axis ranges from -2 to 7. 

The line begins at approximately the point (0, 5) and extends downward and to the right. 

Several points are plotted on the line, including (0, 5), (3, 3), (5, 2), (6, 1). 

An arrowhead is at the far left end of the line and an arrowhead is at the far right end of the line, indicating that the line extends indefinitely in both directions.

Responses

y = 2/3x + 5
y = 2/3x + 5

(y - 3) = -2/3 (x - 3)
(y - 3) = -2/3 (x - 3)

8x + 5y = 3
8x + 5y = 3

y = 5x + 8

1 answer

To find the equation that represents the linear graph, let's analyze the details from the points provided.

  1. The line starts around the point (0, 5) and appears to decrease as x increases, indicating a negative slope.

  2. Analyzing the described points:

    • (0, 5)
    • (3, 3)
    • (5, 2)
    • (6, 1)

Let's calculate the slope (m) between two points, for example, (0, 5) and (3, 3):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 5}{3 - 0} = \frac{-2}{3} \]

Now we have the slope of the line \( m = -\frac{2}{3} \).

  1. Let's use the point-slope form of the equation of a line, which is:

\[ y - y_1 = m(x - x_1) \]

Using the point (3, 3):

\[ y - 3 = -\frac{2}{3}(x - 3) \]

  1. If we rewrite this into slope-intercept form (y = mx + b) or standard form:

\[ y - 3 = -\frac{2}{3}x + 2 \] \[ y = -\frac{2}{3}x + 5 \]

Now, we will compare this with the response options given:

  • \( y = \frac{2}{3}x + 5 \) (not a match)
  • \( (y - 3) = -\frac{2}{3}(x - 3) \) (this is correct)
  • \( 8x + 5y = 3 \) (not a match)
  • \( y = 5x + 8 \) (not a match)

From the analysis, the equation that matches the graph is:

(y - 3) = -\frac{2}{3}(x - 3)

This option accurately represents the equation of the line based on the slope and a point on the line.