# Linear functions

I'm taking Algebra II, and I need help on two types of problems. The numbers may not work out, as I am making these up. 1st problem example: Using the following functions, write a linear functions. f(2) = 3 and f(5) = 4 2nd problem example: Write a solution set for the following equation (I thought you couldn't solve a linear equation?) 2x + 4y = 8 Feel free to use different numbers that actually work in your examples, these problems are just making me scratch my head.

-

First problem. At this level, a "linear function" is one of the form $f(x) = ax+b$ for some $a$ and $b$
If you know that $f(2)=3$ and $f(5)=4$, then by plugging in you get two equations: \begin{align*} 3 &= 2a + b\\ 4 &= 5a + b \end{align*} From these two equations, you should be able to solve for $a$ and $b$, thus finding the function. For example, you can solve for $b$ in the first equation, substitute in the second, and solve the resulting equation for $a$; then plug that to find the value of $b$.
Second Problem. An equation like $2x+4y = 8$ does not have a unique solution, but each value of $x$ gives you a corresponding value of $y$ and vice-versa. The "solution set" of this would be a description of all the values that make the equation true.
For instance, if you had the problem "Write a solution set for $x-3y=4$", you could do the following: given a value of $y$, the value of $x$ has to be $4+3y$. So one way to write the solutions is: $$\bigl\{ (4+3y, y)\,\bigm|\, y\text{ any real number}\bigr\}.$$ For each real number $y$, you get one solution.
Or you could solve for $y$, to get that $y=\frac{x-4}{3}$, and write a solution set as: $$\left\{\left. \left(x, \frac{x-4}{3}\right)\,\right|\, x\text{ any real number}\right\}.$$