Question

On a family outing, Tyler bought 5 cups of hot cocoa and 4 pretzels for

$18.40. Some of his family members would like a second serving, so he
went back to the same food stand and bought another 2 cups of hot cocoa
and 4 pretzels for $11.20.
Here is a system of equations that represent the quantities and constraints
in this situation.
5c + 4p = 18.40
2c + 4p - 11.20

If we add the second equation to the first equation, we have a new equation: 7c + 8p = 29.60
Explain why the same (c, p) pair that is a solution to this new equation.

Answer 1

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Answer:

the properties of equality tell you so

Explanation:

A couple of properties of equality come into play here:

• the addition property of equality
• the substitution property of equality

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.