Use the table to answer the question.

x
y
18 10
6 −3
Write the equation of the linear function represented by the table.

(1 point)
Responses

y=(12/13)x−(111/13)
y equals left parenthesis Start Fraction 12 over 13 End Fraction right parenthesis x minus Start Fraction 111 over 13 End Fraction right parenthesis

y=−19/2
y equals Start Fraction negative 19 over 12 End Fraction

y=(−13/12)x+(7/2)
y equals left parenthesis Start Fraction negative 13 over 12 End Fraction right parenthesis x plus Start Fraction 7 over 2 End Fraction right parenthesis

y=(13/12)x−(19/2)

1 answer

To determine the equation of the linear function represented by the given points, we first find the slope \( m \) using the two points from the table: \( (18, 10) \) and \( (6, -3) \).

The formula for the slope \( m \) between the points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

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

Substituting in the coordinates:

\[ m = \frac{-3 - 10}{6 - 18} = \frac{-13}{-12} = \frac{13}{12} \]

Now that we have the slope, we can use the point-slope form of the linear equation to find the y-intercept. We will use one of the points; let’s use \( (18, 10) \).

The point-slope form of a linear equation is:

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

Plugging in the slope \( m = \frac{13}{12} \) and the point \( (18, 10) \):

\[ y - 10 = \frac{13}{12}(x - 18) \]

Now, we simplify this equation:

\[ y - 10 = \frac{13}{12}x - \frac{13}{12} \cdot 18 \]

Calculating \( \frac{13}{12} \cdot 18 \):

\[ \frac{13 \cdot 18}{12} = \frac{234}{12} = \frac{39}{2} \]

Therefore:

\[ y - 10 = \frac{13}{12}x - \frac{39}{2} \]

Now, add 10 to both sides to isolate \( y \):

\[ y = \frac{13}{12}x - \frac{39}{2} + 10 \]

Converting 10 to a fraction with a common denominator of 2 gives us:

\[ 10 = \frac{20}{2} \]

Thus, we have:

\[ y = \frac{13}{12}x - \frac{39}{2} + \frac{20}{2} \]

This simplifies to:

\[ y = \frac{13}{12}x - \frac{19}{2} \]

Therefore, the equation of the linear function is:

\[ y = \frac{13}{12}x - \frac{19}{2} \]

The correct response is:

y=(13/12)x−(19/2).