Question

3. Solve the following application of a system of linear equations.

Two students in the same Spanish class, Cindy and Ruben, plan to get together after school to make
vocabulary flashcards. Cindy started on the project yesterday and has already made 14 flashcards.
Ruben hasn't started yet so he has none. Since Cindy makes 9 flashcards per minute and Ruben makes
10 flashcards, Ruben will soon have the same number of flashcards.
Which variables will you use and what do they represent?
Write your system of linear equations below.
Solve the system.
How much time will it take for Cindy and Ruben to have the same number of flashcards?
How many flashcards will each student have at that time?

Answer 1

Equations:

`y = 14 + 9 x` --- Cindy

`y = 10 x` --- Ruben

Solution to equation:

Time they have the same amount: 14 minutes

Number of cards they have at that time: 140 flashcards

Explanation:

Solving (a): Variables and what they represent

The variables to use are x and y

Where x represent the minutes and y represents the number of flashcards in x minutes

Solving (b): System of linear equation

Cindy:

`Existing\ Flashcards = 14`

`Additional = 9` per minute

Total number of flashcards (y) in x minutes is:

`y = Existing\ Flashcards + Additional * x`

`y = 14 + 9 * x`

`y = 14 + 9 x`

Ruben:

`Rate = 10` per minute

Total number of flashcards (y) in x minutes is:

`y = Rate * x`

`y = 10 * x`

`y = 10 x`

Solution to Equations:

Time they have the same amount.

To do this, we
`equate`
`both` expressions

i.e.

`10x = 14 + 9x`

Collect Like Terms

`10x - 9x = 14`

`x = 14`

Number of cards they have at that time.

Here, we simply substitute 14 for x in any of the equations.

`y = 10 x`

`y = 10 * 14`

`y = 140`

or

`y = 14 + 9 x`

`y = 14 + 9 * 14`

`y = 14 + 126`

`y = 140`