# What is the equation of the line passing through (1,3), (4,6)?

Mar 20, 2018

$y = x + 2$

#### Explanation:

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

•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)

$\text{let "(x_1,y_1)=(1,3)" and } \left({x}_{2} , {y}_{2}\right) = \left(4 , 6\right)$

$\Rightarrow m = \frac{6 - 3}{4 - 1} = \frac{3}{3} = 1$

$\Rightarrow y = x + b \leftarrow \textcolor{b l u e}{\text{is the partial equation}}$

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

$\text{using "(1,3)" then}$

$3 = 1 + b \Rightarrow b = 3 - 1 = 2$

$\Rightarrow y = x + 2 \leftarrow \textcolor{red}{\text{is the equation of the line}}$

Mar 20, 2018

$y = x + 2$

#### Explanation:

First, we must know what an equation of a line looks like. We write the equation in slope-intercept form:

$y = m x + b$
(The $m$ is the slope, and $b$ is the y-intercept)

Next, find the slope ($m$) of the line by using the formula (y_2-y_1)/ (x_2-x_1):

$\frac{\left(6\right) - \left(3\right)}{\left(4\right) - \left(1\right)}$$=$$\frac{3}{3}$$=$$1$

Next, find the y-intercept ($b$) by using the slope-intercept form equation and substituting $1$ in for $m$ and one of the ordered pairs in for $x$ and $y$:

$\left(3\right) = \left(1\right) \left(1\right) + b$ $\to$ $3 = 1 + b$ $\to$ $2 = b$
-OR-
$\left(6\right) = \left(1\right) \left(4\right) + b$ $\to$ $6 = 4 + b$ $\to$ $2 = b$

Now, we can write the full equation of the line:

$y = x + 2$
(We do not need to put a $1$ in front of $x$ because we know that $1$ times any number equals itself)