### Linear Equations and Graphing

In math and science we are sometimes given a set of data and need to determine an equation to describe the relationship between them. In this lesson we look at how to find the equation of a line given 2 points.

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One real life situation happened when rangers tried to predict how often the geyser Old Faithful would erupt.

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#### Your turn: How do you find the equation of the line through points:

$$\color{#9400d3}{(-3,5)(2,1)}$$

#### Solution:

To solve this problem we will start with a equation for a line. I prefer to use the slope-intercept form: $\color{#9400d3}{y=mx+b}$.

Next we need to find the slope *(m)* of the line.

$$ \color{#9400d3}{m = \frac{\Delta y}{\Delta x} = \frac{5-1}{-3-2} = \frac{4}{-5}}$$

Now we need to fine the *y-intercept*. To do that we will pick a point that is on the line, we could use either $(-3,5)$ or $(2,1)$. It does not matter which pair we use, that other pair will be used in checking our equation. Lets use $\color{#9400d3}{(2,1)}$ to determine the *y-intercept*.

$$

\begin{align*}

y &= \color{#9400d3}{\frac{-4}{5}}x + b\\

\\[1px]

\color{#9400d3}1 &= \frac{-4}{5}*\color{#9400d3}2+b\\

\\[1px]

1 &= \frac{-8}{5}*\color{#9400d3}+b\\

\\[1px]

1 \color{fuchsia}{+\frac{8}{5}} &= \frac{-8}{5}\color{fuchsia}{+\frac{8}{5}}+b\\

\\[1px]

\color{#9400d3}{\frac{13}{5}} &= b\\

\\[1px]

\color{#9400d3}b &= \color{#9400d3}{2\frac{3}{5}}

\end{align*}

$$

Our final step is to add the *y-intercept (b)* into the formula.

$$ \color{#9400d3}{y = {\frac{-4}{5}}x + 2\frac{3}{5}} $$

#### Check:

There two ways we can check this equation.

**1.** by substituting in (-3,5) into the equation and making sure it leads to a true statement (be sure to use the coordinates of the point not used to find the *y-intercept*.)

$$

\begin{align*}

y &= {\frac{-4}{5}}x + 2\frac{3}{5}\\

\\[1px]

\color{#9400d3}5 &= \frac{-4}{5}*\color{#9400d3}{^-3} + 2\frac{3}{5}\\

\\[1px]

5 &= 2\frac{2}{5}+ 2\frac{3}{5}\\

\\[1px]

5 &= 5

\end{align*}

$$

**2.**The other way would be to graph the two original points.

Looking at the graph we can see the *y-intercept* is approximately $2\frac{2}{5}$,

and the slope is $\frac{-4}{5}$.