I can do the following problem in my head, but I want to understand the equations that the Solutions are suggesting. For some reason, they aren't clicking.
What is the cost of a donut?
Solution 1
If a pizza plus a donut costs \$11.00 and a pizza minus a donut costs \$10.50, then we can combine the two equations together to see that two pizzas plus a donut minus a donut equal \$21.50. Therefore, two pizzas must cost \$11.00 + \$10.50 = \$21.50 one pizza costs \$10.75, and one donut costs \$0.25.
I'm not understanding how they arrived at the conclusion in the last sentence, particularly how they got one pizza costs \$10.75.
Solution 2
The first equation, "Pizza + Donut = \$11.00" tells us that, together, the cost of one pizza and one donut is \$11.
The second equation, "Pizza - Donut = \$10.50" tells us that the difference between the the cost of one pizza and one donut is \$10.50. More specifically, it tells that a pizza costs \$10.50 more than a donut: "Pizza = \$10.50 + Donut."
If a pizza costs \$10.50 more than a donut, then a pizza and a donut must cost \$10.50 more than two donuts: "Pizza + Donut = $10.50 + Donut + Donut."
Therefore, "\$11.00 = \$10.50 + Donut + Donut."
And we can see that two donuts must cost \$0.50, and one donut must therefore cost \$0.25.
I start to lose track of what they're saying starting here:
If a pizza costs \$10.50 more than a donut, then a pizza and a donut must cost \$10.50 more than two donuts:
I know this is quite a simple problem, and easy to do in my head, but I'd really like to understand the fundamentals of building these equations for harder problems.
It helps to use symbols for this. So we have $ P+D=x$ and $P-D=y$.
Now add and subtract the two equations to get $2P=x+y$ and $2D=x-y$.
Finally divide both sides of each equation by $2$.