Solving a System of Linear Equations

This post is a bit of a repeat as I took something I did with my grade 10 applied students last year and am using it for my grade 10 academic students this year. We have solved systems of linear equations by graphing and by substitution. Before doing elimination, they normally do this investigation on equivalent systems that takes a very abstract look at why you can multiply equations by a constant and add or subtract equations. In the past students have done the investigation but really did not get anything out of it (other than confusion). I have never liked it, but it was one of those things that all the other teachers were doing, so I assigned it too. No more! This is what we did instead:

The pictures were key to some students' understanding. There were clearly 2 more coffees on the top line and that was the ONLY difference, so those 2 coffees had to account for the difference in price. From there they could work out the price of 1 coffee, then of one doughnut. Then we "translated" it into something more algebraic:

It took a little prompting for them to come up with the idea of subtracting. I asked what they had done with the costs to get them going down the right path.

Next:

I loved hearing a student come up with the idea of doubling the first order. This now gives us an equivalent system (that has meaning) where the number of cookies is the same in both orders so the difference in price is due to the extra latte.

Their homework was to finish this question and do one more that involved multiplying both equations. I really think this method brings meaning to the whole process and will make tomorrow go really smoothly.