# How do you write the point-slope form of the equation below that represents the line that passes through the points (−3, 2) and (2, 1)?

May 27, 2016

$s l o p e = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$
$y = - \frac{1}{5} x + \frac{7}{5}$

#### Explanation:

The slope of the line passing through the two points $\left(- 3 , 2\right)$ and $\left(2 , 1\right)$ is given by

$s l o p e = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

$s l o p e = \frac{1 - 2}{2 - \left(- 3\right)}$

$s l o p e = - \frac{1}{5}$

The equation of the straight line is

$y = a \cdot x + b$

$s l o p e = a = - \frac{1}{5}$

$y = - \frac{1}{5} x + b$

The point $\left(2 , 1\right)$ is a point on the straight line. Plugging the coordinates of this point and the line equation allows us to find the Y-intercept.

$1 = - \frac{1}{5} \times 2 + b$

$b = 1 + \frac{2}{5}$

$b = \frac{7}{5}$

The equation of the straight-line will be

$y = - \frac{1}{5} x + \frac{7}{5}$

To find the X-intercept assign the value 0 to $y$

$0 = - \frac{1}{5} x + \frac{7}{5}$

$\frac{1}{5} x = \frac{7}{5}$

$x = 7$

graph{-1/5*x+7/5 [-9.63, 10.37, -3.4, 6.6]}

May 27, 2016

$y = \frac{- 1 x}{5} + \frac{7}{5}$

#### Explanation:

There are several ways of answering this question:

Method 1 . Use the two points to find the gradient $m$.

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} \text{ }$ Then use one of the points as $\left(x , y\right)$

Substitute $m , x \mathmr{and} y$ into $y = m x + c$ and solve to find $c$.

Now use the values for $m \mathmr{and} c$ in $y = m x + c$ to find the required equation.
This method is fine, but it requires several substitutions and it is easy to lose track of where you are.

Method 2 . Use the two points as $x \mathmr{and} y$ and substitute each into $y = m x + c \text{ }$ This forms two equations which can be solved simultaneously to find m and c.

Method 3 . Use the two points as $\left({x}_{1} , {y}_{1}\right) \mathmr{and} \left({x}_{2} , {y}_{2}\right)$ and substitute into the formula for finding the equation of a line if 2 points are known: $\text{ } \frac{y - {y}_{1}}{x - {x}_{1}} = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

$\text{ } \frac{y - 1}{x - 2} = \frac{2 - 1}{- 3 - 2}$

$\text{ "(y - 1)/(x - 2) = 1/-5 " now cross-multiply}$

$- 5 \left(y - 1\right) = \left(x - 2\right) \text{ solve for y}$

$- 5 y + 5 = x - 2$

$- 5 y \text{ } = x - 2 - 5$

$y = \frac{- 1 x}{5} + \frac{7}{5} \text{ divide by -5}$