Use the table to answer the question.

Question

This table shows the relationship between x and y values that will contribute to linear functions.
x
y

4 38
6 58
Write the equation of the linear function that models the relationship shown in the table.

(1 point)
Responses

y=10x−2
y equals 10 x minus 2

y=10x+2
y equals 10 x plus 2

y=x+34
y equals x plus 34

y=−10x+78
y equals negative 10 x plus 78
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Answer 1

To find the equation of the linear function that relates \( x \) and \( y \) based on the given points \((4, 38)\) and \((6, 58)\), we can use the slope-intercept form of a linear equation, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

First, we need to calculate the slope \( m \):

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

Using the points \((4, 38)\) as \((x_1, y_1)\) and \((6, 58)\) as \((x_2, y_2)\):

\[ m = \frac{58 - 38}{6 - 4} = \frac{20}{2} = 10 \]

Now that we have the slope \( m = 10 \), we can use one of the points to find the y-intercept \( b \). We'll use the point \((4, 38)\):

\[ y = mx + b \]

Substituting \( y = 38 \), \( m = 10 \), and \( x = 4 \):

\[ 38 = 10(4) + b \]

\[ 38 = 40 + b \]

Solving for \( b \):

\[ b = 38 - 40 = -2 \]

Now, we can write the equation of the linear function:

\[ y = 10x - 2 \]

Thus, the correct response is:

y = 10x - 2