Chapter 1: Chapter 1

If the slope of the line \(L_{1}\) is positive, then the slope of a line \(L_{2}\) perpendicular to \(L_{1}\) may be positive or negative.

The lines with equations \(a x+b y+c_{1}=0\) and \(b x-a y+\) \(c_{2}=0\), where $a \neq 0\( and \)b \neq 0$, are perpendicular to each other.

Find the slope of the line that passes through the given pair of points. $$ (a, b) \text { and }(c, d) $$

The accompanying data were compiled by the superintendent of schools in a large metropolitan area. The table shows the average SAT verbal scores of high school seniors during the 5 yr since the district implemented its "back to basics" program. $$ \begin{array}{lccccc} \hline \text { Year, } \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Average } & & & & & \\ \text { Score, } \boldsymbol{y} & 436 & 438 & 428 & 430 & 426 \\ \hline \end{array} $$ a. Determine the equation of the least-squares line for these data. b. Draw a scatter diagram and the least-squares line for these data. c. Use the result obtained in part (a) to predict the average SAT verbal score of high school seniors 2 yr from now \((x=7) .\)

Find the break-even point for the firm whose cost function \(C\) and revenue function \(R\) are given. $$ C(x)=0.2 x+120 ; R(x)=0.4 x $$

If \(L\) is the line with equation \(A x+B y+C=0\), where \(A \neq 0\), then \(L\) crosses the \(x\) -axis at the point \((-C / A, 0)\).

Show that two distinct lines with equations \(a_{1} x+b_{1} y+\) \(c_{1}=0\) and \(a_{2} x+b_{2} y+c_{2}=0\), respectively, are parallel if and only if $a_{1} b_{2}-b_{1} a_{2}=0$. Hint: Write each equation in the slope-intercept form and compare.

Prove that if a line \(L_{1}\) with slope \(m_{1}\) is perpendicular to a line \(L_{2}\) with slope \(m_{2}\), then \(m_{1} m_{2}=-1\). Hint: Refer to the accompanying figure. Show that \(m_{1}=b\) and \(m_{2}=c\). Next, apply the Pythagorean theorem and the distance formula to the triangles \(O A C, O C B\), and \(O B A\) to show that \(1=-b c\).