1

Problem

Find an equation of the linear function given f(-2) = 9 and f(5) = 3.

2

Step

Given functions are

f(-2) = 9 and f(5) = 3

3

Formula

To find the equation of a linear function, we can use the point-slope form of a linear equation: \$y - y_1 = m(x - x_1)\$.

4

Formula

Slope Formula is \$m = (y_2 - y_1) / (x_2 - x_1)\$.

5

Step

Here the points are:

\$ (x_1, y_1)\$ = (- 2, 9) and \$(x_2, y_2) = (5, 3)\$

6

Step

Plug the value in the slope formula:

\$m = (3 - 9) / (5 - (-2)) \$

\$ m = - 6/7 \$

7

Step

Let’s use the point \$(x_1, y_1)\$:

(-2, 9)

8

Step

Now plug the value in the point-slope formula:

\$y - 9 = (- 6/7)(x - (- 2))\$

\$y - 9 = (- 6/7)(x + 2)\$

\$y - 9 = (- 6/7)x - 12/7\$

9

Step

Multiply 7, then simplify:

7(y - 9) = - 6x - 12

7y - 63 = - 6x - 12

\$y = -(6 / 7)x + (51) / 7\$

10

Solution

Therefore, the equation of the linear function is \$f(x) = -(6 / 7)x + (51)/7\$.

11

Sumup

Please summarize steps

Choices

12

Choice-A

This is correct because it done accurately done the calculation by using the formula

Correct \$ -(6/7)x + (51)/7 \$

13

Choice-B

The sign before the \$(51)/7\$​ term is incorrect. It should be positive because the y-intercept we found is positive. So, option B is not correct

Wrong \$ (- 6/7)x - (51)/7\$

14

Choice-C

\$f(x) = (6/7)x + (51)/7\$​, is inaccurate due to its positive slope contrary to the function’s decrease

Wrong \$ (6/7)x + (51)/7\$

15

Choice-D

\$y = (6/7) ​x - (51)/7\$​, has the wrong sign in front of the constant term, it’s negative instead of positive

Wrong \$ (6/7)x - (51)/7\$

16

Answer

Option

A

17

Sumup

Please summarize choices