Richie spent $120 on 12 rose bushes and 2 gardenias, while jim spent it $150 on 10 rose bushes and 10 gardenias. what is the cost of 1 rose bush and 1 gardenia?

With the given information, we can create several equations:

120 = 12x + 2y
150 = 10x + 10y

With x being the number of rose bushes, and y being the number of gardenias.

To find the values of the variables, we can use two methods: Substitution or Elimination

For this case, let us use elimination. To begin, let's be clear that we are going to be adding these equations together. Therefore, in order to get the value of one variable, we must cancel one of them out - it could be x or y, it doesn't matter which one you decide to cancel out. Let's cancel the x out:

We first need to multiply the equations by numbers that would cause the x's to cancel out - and this can be done as follows:

-10(120 = 12x + 2y)
 12(150 = 10x + 10y) => Notice how one of these is negative

Multiply out:

    -1200 = -120x - 20y
 +  1800 =  120x + 120y          => Add these two equations together

600 = 100y

Now we can solve for y:

y = 6

With this value of y known, we can then pick an equation from the beginning of the question, and plug y in to solve for x:

120 = 12x + 2y    =>    120 = 12x + 2(6)

Now we can solve for x:

120 = 12x + 12    =>     108 = 12x

x = 9

So now we know that x = 9, and y = 6.

With rose bushes being x, we now know that the cost of 1 rose bush is $9.
With gardenias being y, we now know that the cost of 1 gardenia is $6.  

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