How do you write out the equation of the line passing through (1,5) (6,15)?

Nov 15, 2016

$y = 2 x + 3$

Explanation:

Use point-slope and slope formula.

We first need to find the slope by finding the change in y over the change in x. This would be $\frac{15 - 5}{6 - 1} = \frac{10}{5}$, which gives us $2$.

Now if we look at point slope formula, it is $y - {y}_{1} = m \left(x - {x}_{1}\right)$ where ${x}_{1}$ and ${y}_{1}$ are the x and y coordinates of a given point. Let's just say we plug in $\left(1 , 5\right)$ for it and the $2$ for $m$ which is slope.

$y - 5 = 2 \left(x - 1\right)$

Add $5$ to both sides.

$y \setminus \cancel{- 5} \setminus \cancel{\setminus \textcolor{\in \mathrm{di} a n red}{+ 5}} = 2 \left(x - 1\right) \setminus \textcolor{\in \mathrm{di} a n red}{+ 5}$

Distribute $2$

$y = \setminus \textcolor{n a v y}{2 x - 2} + 5$

Add $- 2$ and $5$

$y = 2 x + \setminus \textcolor{o l i v e}{3}$

Nov 15, 2016

$y = 2 x + 3$

Apply slope formula

• use formula $m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$ given two points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ (doesn't matter which one comes first)
the slope you get should be $\setminus \textcolor{o l i v e}{m} = \frac{10}{5} = \frac{2}{1} = \setminus \textcolor{o l i v e}{2}$

Apply point-slope formula

• now plug-in one of the points for the formula $y - {y}_{1} = m \left(x - {x}_{1}\right)$,
which will now become $y - {y}_{1} = \setminus \textcolor{o l i v e}{2} \left(x - {x}_{1}\right)$.
here, $\setminus \textcolor{\in \mathrm{di} a n red}{\left({x}_{1} , {y}_{1}\right)}$ can be either of the points.
for easier explanation, used $\setminus \textcolor{\in \mathrm{di} a n red}{\left(1 , 5\right)}$

Working it out

step A $\setminus \textcolor{m a r \infty n}{\rightarrow}$plugging in: $y - \setminus \textcolor{\in \mathrm{di} a n red}{5} = 2 \left(x - \setminus \textcolor{\in \mathrm{di} a n red}{1}\right)$
step B $\setminus \textcolor{m a r \infty n}{\rightarrow}$distributing: $y - 5 = \setminus \textcolor{s e a g r e e n}{2 x - 2}$
step C $\setminus \textcolor{m a r \infty n}{\rightarrow}$$\setminus \textcolor{t e a l}{5}$ added to each side: $y \setminus \cancel{- 5} \setminus \cancel{\setminus \textcolor{t e a l}{+ 5}} = 2 x - 2 \setminus \textcolor{t e a l}{+ 5}$
step D $\setminus \textcolor{m a r \infty n}{\rightarrow}$simplifying like terms: $y = 2 x + \setminus \textcolor{g r e e n}{3}$

• therefore the equation is:
$\setminus \textcolor{c \mathmr{and} n f l o w e r b l u e}{y = 2 x + 3}$