39. Write a linear function $$ f $$ with values $$ f(-3)=5 $$ and $$ f(6)=11 $$.

Question

UpStudy AI · Accepted Answer

We start with the general form of a linear function:
$$
f(x)=ax+b.
$$

Given that $$ f(-3)=5 $$, we have:
$$
a(-3)+b=5 \quad \Rightarrow \quad -3a+b=5.
$$

Given that $$ f(6)=11 $$, we have:
$$
a(6)+b=11 \quad \Rightarrow \quad 6a+b=11.
$$

Subtract the first equation from the second:
$$
(6a+b) - (-3a+b)=11-5 \quad \Rightarrow \quad 9a=6.
$$

Solve for $$ a $$:
$$
a=\frac{6}{9}=\frac{2}{3}.
$$

Substitute $$ a=\frac{2}{3} $$ into the equation $$-3a+b=5$$:
$$
-3\left(\frac{2}{3}\right)+b=5 \quad \Rightarrow \quad -2+b=5.
$$

Solve for $$ b $$:
$$
b=5+2=7.
$$

Thus, the linear function is:
$$
f(x)=\frac{2}{3}x+7.
$$
The linear function is $$ f(x) = \frac{2}{3}x + 7 $$.

39. Write a linear function \( f \) with values \( f(-3)=5 \) and \( f(6)=11 \).