# How do you write the standard form of a line given (4,3) and (7, -2)?

Jun 2, 2018

$y = - \frac{5}{3} \cdot x + \frac{29}{3}$

#### Explanation:

The slope can be computed as
$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{- 2 - 3}{7 - 4} = - \frac{5}{3}$
so we get
$y = - \frac{5}{3} x + n$
substituting
$x = 4 , y = 3$
we get
$3 + \frac{20}{3} = n$ so $n = \frac{29}{3}$

Jun 2, 2018

$5 x + 3 y = 29$

#### Explanation:

$\text{the equation of a line in "color(blue)"standard form}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{A x + B y = C} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{where A is a positive integer and B, C are integers}$

$\text{to begin obtain the equation in "color(blue)"slope-intercept form}$

•color(white)(x)y=mx+b

$\text{where m is the slope and b the y-intercept}$

$\text{to calculate m use the "color(blue)"gradient formula}$

•color(white)(x)m=(y_2-y_1)/(x_2-x_1)

$\text{let "(x_1,y_1)=(4,3)" and } \left({x}_{2} , {y}_{2}\right) = \left(7 , - 2\right)$

$m = \frac{- 2 - 3}{7 - 4} = \frac{- 5}{3} = - \frac{5}{3}$

$y = - \frac{5}{3} x + b \leftarrow \textcolor{b l u e}{\text{is the partial equation}}$

$\text{to find b substitute either of the 2 given points into the}$
$\text{partial equation}$

$\text{using "(4,3)" then}$

$3 = - \frac{20}{3} + b \Rightarrow b = \frac{9}{3} + \frac{20}{3} = \frac{29}{3}$

$y = - \frac{5}{3} x + \frac{29}{3} \leftarrow \textcolor{red}{\text{in slope-intercept form}}$

$\text{multiply all terms by 3}$

$3 y = - 5 x + 29$

$\text{add "5x" to both sides}$

$5 x + 3 y = 29 \leftarrow \textcolor{red}{\text{in standard form}}$

Jun 19, 2018

color(green)(5x + 3y = 29 " is the standard form"

#### Explanation:

color(crimson)("Standard form of linear equation is " ax + by = c

"Given points are " (x_1, y_1) = 4,3), (x_2,y_2) = (7,-2)

Knowing two points on a line, we can form the equation using the formula,

$\frac{y - {y}_{1}}{{y}_{2} - {y}_{1}} = \frac{x - {x}_{1}}{{x}_{2} - {x}_{1}}$

$\frac{y - 3}{- 2 - 3} = \frac{x - 4}{7 - 4}$

$\frac{y - 3}{-} 5 = \frac{x - 4}{3}$

$3 y - 9 = - 5 x + 20 , \text{ cross-multiplying}$

color(green)(5x + 3y = 29 " is the standard form"

graph{-(5/3)x + (29/3) [-10, 10, -5, 5]}