# How do you get these two equations into one equation?

## Find an exponential function that passes through each pair of points. Write your answer like y=a(b)^x The ordered pairs are: (1,7.5) and (3,16.875) What are the equations to both of them? I want to know if I did it right.

Feb 6, 2018

$y = 5 \cdot {1.5}^{x}$

#### Explanation:

The problem tells us that our function will look like this:

$y = a \cdot {b}^{x}$

So, if we can solve for $a$ and $b$, we will get our equation. We know that the equation must satisfy both of the given points, so if we plug in both pairs of coordinates, we will get two true equations in terms of $a$ and $b$ that we can use to solve for $a$ and $b$.

So, let's plug in the two coordinates and see what we get:

First, the point $\left(\textcolor{red}{1} , \textcolor{b l u e}{7.5}\right)$

$\textcolor{b l u e}{7.5} = a \cdot {b}^{\textcolor{red}{1}}$
$\textcolor{b l u e}{7.5} = a b$

Next, the point $\left(\textcolor{red}{3} , \textcolor{b l u e}{16.875}\right)$

$\textcolor{b l u e}{16.875} = a \cdot {b}^{\textcolor{red}{3}}$
$\textcolor{b l u e}{16.875} = a {b}^{3}$

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This gives us a system of equations in terms of $a$ and $b$, which we can solve:

$7.5 = a b \text{ "" "and" "" } 16.875 = a {b}^{3}$

Since $16.875$ is equal to $a {b}^{3}$, and $7.5$ is equal to $a b$, if we divide $16.875$ by $7.5$, it should be the same as what we get when we divide $a {b}^{3}$ by $a b$. Therefore:

$\frac{16.875}{7.5} = \frac{a {b}^{3}}{a b}$

$2.25 = \frac{\cancel{a} \cdot \cancel{b} \cdot {b}^{2}}{\cancel{a} \cdot \cancel{b}}$

$2.25 = {b}^{2}$

$\pm 1.5 = b$

And since $b$ is the base of the exponential equation, it cannot be negative since ${b}^{x}$ is not continuous for any $x$ if $b$ is negative. Therefore:

$b = 1.5$

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Now that we have solved for $b$, we can use one of our equations from earlier to solve for $a$ as well. Remember that $b = \textcolor{\mathmr{and} a n \ge}{1.5}$.

$7.5 = a b$

$7.5 = a \left(\textcolor{\mathmr{and} a n \ge}{1.5}\right)$

$5 = a$

So $a = 5$ and $b = 1.5$. This means that the exponential equation is:

$y = 5 \cdot {1.5}^{x}$