# What is the equation of a line that goes through (2,2) and (3,6)?

Mar 6, 2018

$y = 4 x - 6$

#### Explanation:

Step 1: You have two points in your question: $\left(2 , 2\right)$ and $\left(3 , 6\right)$. What you need to do, is use the slope formula. The slope formula is

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

Step 2: So let's look at the first point in the question. $\left(2 , 2\right)$ is (x_1,y_1. That means that $2 = {x}_{1}$ and $2 = {y}_{1}$. Now, let's do the same thing with the Second point $\left(3 , 6\right)$. Here $3 = {x}_{2}$ and $6 = {y}_{2}$.

Step 3: Let's plug those numbers into our equation. So we have

$m = \frac{6 - 2}{3 - 2} = \frac{4}{1}$

That gives us an answer of $4$! And the slope is represented by the letter $m$.

Step 4: Now let's use our equation of a line formula. That slope-intercept equation of a line is

$y = m x + b$

Step 5: Plug in one of the points: either $\left(2 , 2\right)$ or $\left(3 , 6\right)$ into $y = m x + b$. Thus, you have

$6 = m 3 + b$

Or you have

$2 = m 2 + b$

Step 6: You have $6 = m 3 + b$ OR you have $2 = m 2 + b$. We also found our m earlier in step 3. So if you plug in the $m$, you have

$6 = 4 \left(3\right) + b \text{ " or " } 2 = 4 \left(2\right) + b$

Step 7: Multiply the $4$ and $3$ together. That gives you $12$. So you have

$6 = 12 + b$

Subtract the $12$ from both sides and you now have

$- 6 = b$

OR

Multiply $4$ and $2$ together. That gives you $8$. So you have

$2 = 8 + b$

Subtract $8$ from both sides and you now have

$- 6 = b$

Step 8: So you have found $b$ and $m$! That was the goal! So your equation of a line that goes through $\left(2 , 2\right)$ and $\left(3 , 6\right)$ is

$y = 4 x - 6$