Chapter 1.1, Problem 5E

(a)

To determine

To plot: The points $A\left(1,-2\right)$ and $B\left(2,1\right)$ .

Explanation of Solution

Given information: The two points for a line $L$ are $A\left(1,-2\right)$ and $B\left(2,1\right)$ .

Graph:

Plot the points $A\left(1,-2\right)$ and $B\left(2,1\right)$ on the coordinate plane as shown below.

  

Figure (1)

Interpretation: From the above graph it can be seen that the point A is 2 units downward from x -axis and point B is 1 unit upward from x -axis.

(b)

To determine

To find: The slope of line $L$ .

Answer to Problem 5E

The slope of line $L$ is equal to $3$ .

Explanation of Solution

Given information: The two points for a line $L$ are $A\left(1,-2\right)$ and $B\left(2,1\right)$ .

Formula used: The formula for the slope of line determined by points $A\left(x_1,y_1\right)$ and $B\left(x_2,y_2\right)$ is:

  $m=\frac{y_2-y_1}{x_2-x_1}$

Calculation:

Substitute $1$ , $2$ for $x_1$ , $x_2$ and $-2$ , $1$ for $y_1$ , $y_2$ respectively in the formula for slope.

  $\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{1-\left(-2\right)}{2-1}\\ =\frac{3}{1}\\ =3\end{array}$

Therefore, the slope of line $L$ is equal to $3$ .

(c)

To determine

To graph: The line $L$ passes through the points $A\left(1,-2\right)$ and $B\left(2,1\right)$ .

Explanation of Solution

Given information: The two points for a line $L$ are $A\left(1,-2\right)$ and $B\left(2,1\right)$ .

Graph:

The equation of line with slope $m$ and passing through the point $\left(x_1,y_1\right)$ is given by,

  $y=m\left(x-x_1\right)+y_1$

As calculated in part (b), the slope of the line $L$ is $3$ and line passes through $\left(1,-2\right)$ .

Substitute $1$ , $-2$ for $x_1$ , $y_1$ and $3$ for $m$ in the general equation for the equation of line $L$ .

  $\begin{array}{c}y=m\left(x-x_1\right)+y_1\\ =3\left(x-1\right)+\left(-2\right)\\ =3x-3-2\\ =3x-5\end{array}$

So, the equation of the line $L$ is $y=3x-5$ .

Draw the graph of the line $L$ as shown below on the coordinate plane.

  

Figure (2)

Interpretation: From the above graph it can be interpreted that the y -intercept of the equation is $\left(0,5\right)$ .