Exercise 27 Page 98

Equations in standard form are written in a specific format. In this form, $A,$ $B,$ and $C$ are integers and $A$ and $B$ cannot both be zero. In this case, we haven't been given an equation, so we will need to use the given point and slope to write one.

Writing an Equation

The quickest way to write an equation, with a given point and slope, is by using the point-slope form. An equation in point-slope form follows a specific format.

$$\begin{array}{c}y-y_1=m\left(x-x_1\right)\end{array}$$

In this form, $m$ is the slope and the point $(x_1,y_1)$ lies on the graph of the line. Using the point $(2,6)$ as our point and the slope $2,$ we have everything needed to form a point-slope form equation.

$$\begin{array}{c}y-6=2\left(x-2\right)\end{array}$$

Note that there are infinitely many correct ways to write this equation in point-slope form, as long as we use a point that lies on the line.

Rearranging to Standard Form

We will use the Properties of Equality to rewrite the given equation in the standard form.

$y-6=2(x-2)$

Distr

Distribute $2$

$y-6=2x-4$

AddEqn

$\text{LHS}+6=\text{RHS}+6$

$y=2x+2$

SubEqn

$\text{LHS}-2x=\text{RHS}-2x$

$\text{-}2x+y=2$

Let's compare this result with the general standard form equation.

$$\begin{array}{rl}Ax+By& =C\\ \text{-}2x+1y& =2\end{array}$$