Equations of Linear Functions Quick Check

Question

3 of 53 of 5 Items
Question
Use the table to answer the 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 plus 2

\(y=-10x+78\)
y equals negative 10 x plus 78

\(y=10x-2\)
y equals 10 x minus 2

\(y=x+34\)
y equals x plus 34
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Answer 1

To find the equation of the linear function that models the relationship shown in the table, we can use the two points given: (4, 38) and (6, 58). The first step is to calculate the slope (m) using the slope formula:

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

Substituting the values from the table:

• \(y_1 = 38\)
• \(y_2 = 58\)
• \(x_1 = 4\)
• \(x_2 = 6\)

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

Now that we have the slope \(m = 10\), we can use the point-slope form of a linear equation, which is:

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

Using the point (4, 38):

\[ y - 38 = 10(x - 4) \]

Simplifying this:

\[ y - 38 = 10x - 40 \] \[ y = 10x - 2 \]

Thus, the equation of the linear function that models the relationship shown in the table is:

\[ y = 10x - 2 \]

The correct response is: \(y = 10x - 2\)