Question

Tavon has a gift card for $70 that loses $3 for each​ 30-day period it is not used. He has another gift card for $ 60 that loses $ 2.50 for each​ 30-day period it is not used. a. Write and solve an equation for the number of​ 30-day periods until the value of the gift cards will be equal. b. What will the value of each card be when they have equal​ value? What will be the equation?

Answer 1

A.
`70-3x=60-2.50x`

Solution:
`x=20`

B. The value of each card after twenty 30-day periods will be $10

Explanation:

Let x be the number of​ 30-day periods.

First gift card:

Amount = $70

Lose for one 30-day period = $3

Lose for x 30-day periods = $3x

Left = $(70-3x)

Second gift card:

Amount = $60

Lose for one 30-day period = $2.50

Lose for x 30-day periods = $2.50x

Left = $(60-2.50x)

If the value of the gift cards are equal, then

`70-3x=60-2.50x`

Solve this equation. First, separate variables and numbers into different sides:

`-3x+2.50x=60-70\\ \\-0.50x=-10\\ \\0.5x=10`

Multiply by 10:

`5x=100\\ \\x=(100)/(5)=20`

If x = 20, then

`70-3x=70-3\cdot 20=70-60=10\\ \\60-2.50\cdot 20=60-50=10`

Answer 2

a. Equation:
`70-3x=60-2.50x`

The value of gift cards will be equal after 20 30-day-periods.

b. The value of each gift card when they have equal values will be $10.

Equation:
`y=70-3(20)`

Explanation:

Let be "x" the number of 30-day periods.

a. We know that Tavon has a gift card for $70 and it loses $3 for each​ 30-day period it is not used. This can be expressed as:

`70-3x`

The other gift card for $ 60 loses $ 2.50 for each​ 30-day period it is not used. Then:

`60-2.50x`

When the value of the gift cards are equal:

`70-3x=60-2.50x`

Solve for "x":

`70-3x=60-2.50x\\\\70-60=-2.50x+3x\\\\10=0.5x\\\\x=20`

Therefore, the value of the gift cards will be equal after 20 30-day-periods.

b. Let be "y" the value of each card (in dollars) when they have equal value. Choose one the expressions shown above. Let's pick the first expression
`70-3x`. Then:

`y=70-3x`

Substitute
`x=20` into the equation, then:

`y=70-3(20)`

Evaluate it in order to find what will be the value of each card when they have equal value:

`y=70-3(20)\\\\y=10\$`