Leila makes $4 per hour and has $30 in her savings account. Mark makes $5 an hour and has $5 in his savings account.

Question

After how many hours will they have the same amount of money?

Answer 1

To find out after how many hours Leila and Mark will have the same amount of money, we can set up equations based on their earnings and initial savings.

Let \( h \) be the number of hours they work.

Leila's total amount of money after \( h \) hours is: \[ \text{Leila's Money} = 30 + 4h \]

Mark's total amount of money after \( h \) hours is: \[ \text{Mark's Money} = 5 + 5h \]

We want to find the value of \( h \) when their total amounts are equal: \[ 30 + 4h = 5 + 5h \]

Now, let's solve for \( h \):

1. Subtract \( 4h \) from both sides: \[ 30 = 5 + 5h - 4h \] Simplifying gives: \[ 30 = 5 + h \]
2. Subtract 5 from both sides: \[ 30 - 5 = h \] Thus: \[ h = 25 \]

So, Leila and Mark will have the same amount of money after 25 hours.

Answer 2

a recreation center is offering special prices on its pool passes and gym memberships for the summer. on the first day of the offering, a family paid $96 for 4 pool passes and 2 gym memberships. later that day, an individual bought a pool pass for herself, a pool pass for a friend, and 1 gym membership. she paid $72

write ten things you know about this

Answer 3

Based on the information provided about the recreation center's special prices on pool passes and gym memberships, here are ten things we can deduce:

1. Price for Pool Passes and Gym Memberships: There are two items being sold: pool passes and gym memberships. Their individual prices are not known yet.

2. First Transaction Details: A family paid a total of $96 for 4 pool passes and 2 gym memberships.

3. Second Transaction Details: An individual paid a total of $72 for 2 pool passes and 1 gym membership.

4. Linear Equations Setup: We can set up a system of linear equations to represent the two transactions based on the prices of pool passes (let's call it \( p \)) and gym memberships (let's call it \( g \)):

   • From the first transaction: \( 4p + 2g = 96 \)
   • From the second transaction: \( 2p + g = 72 \)

5. Two Equations with Two Unknowns: We have two equations with two variables, allowing us to solve for the values of \( p \) and \( g \).

6. Potential Discounts: Since there are special prices for the summer, we might expect the prices to be lower than regular rates, but we can't determine that without additional information.

7. Family vs. Individual Purchases: The first transaction involves a family buying in bulk (4 pool passes and 2 gym memberships), while the second transaction suggests an individual purchase, indicating possible differences in pricing models or discounts for bulk purchases.

8. Total Revenue Calculated: The total revenue collected from both transactions is $96 + $72 = $168.

9. Proportionality: Since the first family purchased twice as many gym memberships as the second transaction, it might suggest a trend in how prices are structured (potentially per item).

10. Problem-Solving Opportunity: This scenario presents an opportunity to introduce concepts of algebra, particularly solving systems of equations, to find out the exact prices of the pool passes and gym memberships.

With these points, we can approach the problem methodically to find the actual prices of the pool passes and gym memberships if needed. Would you like to proceed with solving the equations to find the exact prices?

Answer 4

write one more

Answer 5

11. Comparison of Purchases: The family spent $96 for 4 pool passes and 2 gym memberships, while the individual spent $72 for only 2 pool passes and 1 gym membership. This comparison shows that the family paid more in total, suggesting that the overall cost of buying pool passes and gym memberships might offer savings through family or bulk pricing.

Answer 6

What terms can eliminate themselves?
For example:
3x + _____ would eliminate to 0.

Answer 7

To eliminate a term from an equation, you can use its additive inverse. Specifically, if you have a term such as \( 3x \), you can eliminate it by adding \( -3x \) to it.

Thus, the expression would be:

\[ 3x + (-3x) = 0 \]

In general terms, any mathematical expression \( a + (-a) = 0 \) will eliminate itself.

Here are a few examples of terms that can eliminate themselves:

1. \( x + (-x) = 0 \)
2. \( 5y + (-5y) = 0 \)
3. \( 10z + (-10z) = 0 \)
4. \( 2a + (-2a) = 0 \)
5. \( 7 + (-7) = 0 \)

In all these cases, adding the term and its negative will result in zero, effectively eliminating it from the equation.

Answer 8

How many equations do we need for solving if we have 2 variables?

Answer 9

To solve a system of equations with 2 variables, you generally need two equations. This is because:

1. A Single Equation: One equation can represent a line in a two-dimensional space, and it has infinitely many solutions (points on that line).

2. Two Equations: When you have two equations, you can find their intersection point, which represents the unique solution to the system (the exact values of both variables that satisfy both equations).

So, to summarize, for 2 variables, you need two equations to find a unique solution.

Here are the types of outcomes you can expect:

• One Unique Solution: The two equations intersect at one point.
• No Solution: The two equations are parallel and do not intersect.
• Infinitely Many Solutions: The two equations represent the same line (are dependent).

Let me know if you need further clarification or examples!