# Question #c9986

May 6, 2017

$y = 2 x - 5$

#### Explanation:

Let’s start from a very simple equation:

$y = x$

From the graph of this equation, you can tell that it has a slope of $1$ because it rises $1$ for every $1$ we run.

But we need a slope of $2$. This means that we need a line that rises $2$ for every $1$ you run. We have to double the steepness of our line.

To double the $y$ for every $x$ that we run, we change our equation to:

$y = 2 x$

You can graph this equation to verify that it rises $2$ for every $1$ we run.

But this line does not pass through point $\left(3 , 1\right)$. We know that because, when $x$ is $3$, $y$ is $6$ (not $1$).

By subtracting $5$, we will make every $y$ coordinate of our line go down $5$ units. And we do want that. So we try the equation:

$y = 2 x - 5$

Let’s try this equation with $x = 2$, $x = 3$, and $x = 4$.

Say $x = 2$. Then $y = 2 \left(2\right) - 5 = 4 - 5 = - 1$

Say $x = 3$. Then $y = 2 \left(3\right) - 5 = 6 - 5 = 1$

Say $x = 4$. Then $y = 2 \left(4\right) - 5 = 8 - 5 = 3$

From this, we notice two things. The first one is that our line rises $2$ for every $1$ we run. So it still has a slope of $2$. The second one is that when $x = 3$, $y = 1$. This tells us that our line passes through $\left(3 , 1\right)$.

$y = 2 x - 5$