Question

Write an equation for the linear function satisfying the given conditions f(-3)=5 and f(6)=-2

Answer 1

The two points (-3,5) and (6,-2) are on this line. Using the slope formula, we get
m = (y2-y1)/(x2-x1)
m = (-2-5)/(6-(-3))
m = (-2-5)/(6+3)
m = -7/9
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Now use this slope value, and the point (x,y) = (-3,5), to find the y intercept b

y = mx+b
5 = (-7/9)(-3)+b
5 = 7/3+b
5-7/3 = 7/3+b-7/3
8/3 = b
b = 8/3
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We go from y = mx+b to y = (-7/9)x + 8/3

So the linear function is $f(x) = -\frac{7}{9}x+\frac{8}{3}$ which is the final answer

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Check:
Plug in x = -3
$f(x) = -\frac{7}{9}x+\frac{8}{3}$

$f(-3) = -\frac{7}{9}(-3)+\frac{8}{3}$

$f(-3) = \frac{7}{3}+\frac{8}{3}$

$f(-3) = \frac{7+8}{3}$

$f(-3) = \frac{15}{3}$

$f(-3) = 5$

So that checks out. 

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Plug in x = 6

$f(x) = -\frac{7}{9}x+\frac{8}{3}$

$f(6) = -\frac{7}{9}(6)+\frac{8}{3}$

$f(6) = -\frac{14}{3}+\frac{8}{3}$

$f(6) = \frac{-14+8}{3}$

$f(6) = \frac{-6}{3}$

$f(6) = -2$

and that checks out at as well. The answer has been fully confirmed.