Question

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?

Answer 1

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 `2y`. So, `x` is 3 less than `2y`. As an equation, it looks like
`x=2y-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=2y-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*(x-6)-3`
Open the parenthesis:
`x=2x-12-3`
`x=2x-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*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.