Chapter 1.1, Problem 21E

To determine

To find: The general linear equation for the line through the two points $\left(0,0\right)$ and $\left(2,3\right)$ .

Answer to Problem 21E

The general linear equation for the line through the two points $\left(0,0\right)$ and $\left(2,3\right)$ is $3x-2y=0$ .

Explanation of Solution

Given information: The line passes through the two points $\left(0,0\right)$ and $\left(2,3\right)$ .

Formula used: The general linear equation of a line in terms of x and y is:

  $Ax+By=C$

Calculation:

The formula for the slope of a line that passes through points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is:

  $m=\frac{y_2-y_1}{x_2-x_1}$

Substitute 0 for $x_1$ , 0 for $y_1$ , 2 for $x_2$ and 3 for $y_2$ in the formula of slope.

  $\begin{array}{c}m=\frac{3-0}{2-0}\\ =\frac{3}{2}\end{array}$

The slope-intercept form of an equation of line with slope m and point $\left(x_1,y_1\right)$ is given by,

  $y=m\left(x-x_1\right)+y_1$

Substitute 0 for $x_1$ , 0 for $y_1$ and $\frac{3}{2}$ for m in the above formula.

  $\begin{array}{c}y=\frac{3}{2}\left(x-0\right)+0\\ y=\frac{3}{2}x\\ 2y=3x\\ 3x-2y=0\end{array}$

Therefore, the general linear equation for the line through the two points $\left(0,0\right)$ and $\left(2,3\right)$ is $3x-2y=0$ .