# Rachel and Kyle both collect geodes. Rachel has 3 less than twice the number of geodes Kyle has. Kyle has 6 fewer geodes than Rachel.How do you write a system of equations to represent this situation and solve?

May 16, 2015

Problems like this are solved using a system of equations. To create this system, look at each sentence and try to reflect it in the equation.

Assume, Rachel has $x$ geodes and Kyle has $y$ geodes. We have two unknowns, which means we need two independent equations.

Let's transform into an equation the first statement about these quantities: "Rachel has 3 less than twice the number of geodes Kyle has." What it says is that $x$ is 3 less than double $y$. Double $y$ is $2 y$. So, $x$ is 3 less than $2 y$. As an equation, it looks like
$x = 2 y - 3$

The next statement is "Kyle has 6 fewer geodes than Rachel." So, $y$ is 6 fewer than $x$. That means:
$y = x - 6$.

So , we have a system of equations:
$x = 2 y - 3$
$y = x - 6$

The easiest way to solve this system is to substitute $y$ from the second equation into the first to have only one equation with one variable:
$x = 2 \cdot \left(x - 6\right) - 3$
Open the parenthesis:
$x = 2 x - 12 - 3$
$x = 2 x - 15$
Add $15 - x$ to both sides to separate $x$ from numeric constants:
$15 = x$
So, the $x = 15$.
The value of $y$ can be determined from the second equation:
$y = x - 6 = 15 - 6 = 9$

So, Rachel has 15 geodes, Kyle has 9 geodes.

Checking step is very much desirable.
(a) Check "Rachel has 3 less than twice the number of geodes Kyle has."
Indeed, twice as Kyle has is $9 \cdot 2 = 18$ geodes.
Rachel's 15 geodes are 3 less than 18.
(b) Check "Kyle has 6 fewer geodes than Rachel".
Indeed, Kyle's 9 geodes are 6 less than Rachel's 16.

This confirms the correctness of the obtained solution.