Alexandra purchases two donuts and three cookies at the donut shop and it's charged $3.30. Brianna purchases five donuts and two cookies at the same shop for $4.95. All the donuts have the same price and all the cookies have the same price. What is the cost in dollars of each donut?

Given:

Cost of two donuts and three cookies = $3.30

Cost of five donuts and two cookies = $4.95

To find:

The cost of each donut.

Solution:

Let x be the cost of each donut and y be the cost of each cookie.

Cost of two donuts and three cookies = $3.30

         ...(i)

Cost of five donuts and two cookies = $4.95

         ...(ii)

Multiply equation (i) by 2.

         ...(iii)

Multiply equation (ii) by 3.

     ...(iv)

Subtracting (iii) from (iv), we get

Divide both sides by 11.

Therefore, the cost of each donut is $0.75.

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  the properties of equality tell you so

Step-by-step explanation:

A couple of properties of equality come into play here:

Each of the properties of equality tells you an operation you can perform on an equation without changing the values of the variables that satisfy that equation.

If we start with the equation ...

  5c + 4p = 18.40

and we add 11.20 to both sides, we have not changed the values of the variables that satisfy this equation. This is the addition property of equality.

  (5c +4p) +11.20 = (18.40) +11.20

The substitution property of equality says I can substitute anything for its equal. The second given equation tells me that 2c +4p = 11.20. This means I can use 2c+4p wherever 11.20 might be found without changing any variable values. In the equation we got from addition, we can now make this substitution, and we have ...

  (5c +4p) +(2c +4p) = (18.40) +(11.20)

  7c +8p = 29.60 . . . . . . . collect terms

We obtained this equation using properties of equality that guarantee solutions to these equations are not altered by the process.

*Additional comment

Technically, by doing the "collect terms" step, we have made the assumption that the values of c and p that satisfy the first equation are the same as the values of c and p that satisfy the second equation. This is usually the case we're interested in.