Chapter 1.1, Problem 43E

Consider the linear system
$\left|\begin{array}{c}x+y=1\\ x+\frac{t}{2}y=t\end{array}\right|$ ,
wheret is a nonzero constant.
a. Determine the x- and -intercepts of the lines $x+y=1$ and $x+\left(t/2\right)y=t$ ; sketch these lines. Forwhich values of the constant t do these lines intersect? For these values of t, the point of intersection $\left(x,y\right)$ depends on the choice of the constant t; thatis, we can consider x and y as functions of t. Drawrough sketches of these functions.

Explain briefly how you found these graphs.Argue geometrically, without solving the system algebraically.
b. Now solve the system algebraically. Verify that thegraphs you sketched in part (a) are compatible withyour algebraic solution.