
Tavon has a gift card for $ 90 that loses $ 4 for each​ 30-day period it is not used. He has another gift card for $ 80 that loses $ 3.50 for each​ 30-day period it is not used.  If x is the number of​ 30-day periods, then the equation 
if x is  the number of​ 30-day periods until the values of the gift cards will be equal.


Let the value of Tavon's two gift cards after $x$ 30-day periods be represented by two linear equations.

### First Gift Card:
- The initial value is $90.
- It loses $4 every 30-day period.
- The equation for the remaining value of the first card after $x$ periods is:
\[
90 - 4x
\]

### Second Gift Card:
- The initial value is $80.
- It loses $3.50 every 30-day period.
- The equation for the remaining value of the second card after $x$ periods is:
\[
80 - 3.50x
\]

Now, to find when the values of the gift cards are equal, set the two expressions equal to each other:
\[
90 - 4x = 80 - 3.50x
\]

### Solving the equation:
1. Subtract $80$ from both sides:
\[
90 - 80 - 4x = -3.50x
\]
\[
10 - 4x = -3.50x
\]

2. Add $4x$ to both sides:
\[
10 = 0.50x
\]

3. Solve for $x$:
\[
x = \frac{10}{0.50} = 20
\]

Thus, the values of the gift cards will be equal after 20 30-day periods.

Would you like more details on how to interpret this result, or do you have any other questions?

### Related Questions:
1. What will the value of each gift card be after 20 periods?
2. How much will be left on each card after 10 periods?
3. How long will it take for both gift cards to have a value of $0?
4. What is the total amount lost from both cards after 5 periods?
5. What if the cards had different initial values? How would that change the equation?

### Tip:
In problems involving equal values over time, setting up and solving an equation often involves equating the two expressions and solving for the variable, usually by simplifying step by step.

Tavon has a gift card for $90 that loses $4 for each​ 30-day period it is not used. He has another gift card for $80 that loses $3.50 for each​ 30-day period it is not used. If x is the number of​ 30-day periods, then the equation if x is the number of​ 30-day periods until the values of the gift cards will be equal.

Solve a real-world problem involving linear equations where two gift cards, starting with different initial values, lose value over time. Learn how to set up and solve for the number of 30-day periods after which the values of the two gift cards will be equal. Ideal for students in Grades 6-8.

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