Write a system of equations to match each of these problems. Then solve the equations using the method... Substitution, or linear combinations.
1. Maria started her savings account with $105. She adds $10 to it each month, including the first month. Jana Rae started her account with $0. She adds $25 to it each month. After how many months will the girls have the same amound in their accounts? How much will they have?
2. The video store has a basic charge for a 3-day rental and a different per-day late fee. Nate was one day late and paid $5.50 in all for his video. Gina was four days late and paid $10.00 in all for her video. What is the Basic Fee and the Per-Day Late Fee that the store charges?
A real story of her savings account with $105 she adds $10 to each month including the first generation started her account with zero dollars she adds $25 to eight each month after how many months will the girls have the same amount in the account how much will they have
1. Let's define some variables:
- Let's call the number of months that have passed "x".
- Let's call Maria's total amount of money in her account after "x" months "M".
- Let's call Jana Rae's total amount of money in her account after "x" months "J".
Now, let's set up the equations:
- For Maria, her initial amount is $105, and she adds $10 every month. So, her equation is M = 105 + 10x.
- For Jana Rae, her initial amount is $0, and she adds $25 every month. So, her equation is J = 0 + 25x.
We want to find the number of months when the girls have the same amount, so we set M = J and solve for x:
105 + 10x = 25x.
To solve this equation, we can use the method of linear combinations.
Subtracting 10x from both sides gives us:
105 = 15x.
Dividing both sides by 15 gives us:
x = 7.
Therefore, after 7 months, Maria and Jana Rae will have the same amount in their accounts. To find out how much they will have, we substitute x = 7 into either equation. Let's use Maria's equation:
M = 105 + 10x = 105 + 10(7) = 105 + 70 = $175.
So, after 7 months, Maria and Jana Rae will each have $175 in their accounts.
2. Let's define some variables:
- Let's call the basic fee for a 3-day rental "B".
- Let's call the per-day late fee "L".
- Let's call the number of days Nate was late "n".
- Let's call the total amount Nate paid for his video "N".
- Let's call the number of days Gina was late "g".
- Let's call the total amount Gina paid for her video "G".
Now, let's set up the equations:
- For Nate, his equation is N = B + L(n+1), where (n+1) includes the additional day Nate was late.
- For Gina, her equation is G = B + L(g+1), where (g+1) includes the additional 4 days Gina was late.
We want to find the values of B and L, so we'll set up a system of equations using the given information:
Equation 1: N = B + L(n+1) and Equation 2: G = B + L(g+1).
Using the given information, we substitute the values of N, n, G, and g into the equations:
Equation 1: 5.50 = B + L(1) and Equation 2: 10.00 = B + L(4).
To solve this system of equations, we can use substitution or linear combinations. Let's use substitution:
From Equation 1, we have: B = 5.50 - L.
Substituting B into Equation 2:
10.00 = (5.50 - L) + L(4).
Simplifying the equation:
10.00 = 5.50 + 4L.
Subtracting 5.50 from both sides:
4.50 = 4L.
Dividing both sides by 4:
L = 1.125.
Substituting the value of L into Equation 1 to find B:
B = 5.50 - L = 5.50 - 1.125 = 4.375.
Therefore, the basic fee charged by the store is $4.375 and the per-day late fee is $1.125.
1. To solve the first problem using substitution or linear combinations, we need to create a system of equations that represents the situation of Maria and Jana Rae's savings accounts.
Let's use the variables "M" to represent Maria's account balance and "J" to represent Jana Rae's account balance at any given month.
For the first equation, we can say that Maria's account balance (M) is equal to her starting amount ($105) plus $10 multiplied by the number of months (t) she has been adding money:
M = 105 + 10t
For the second equation, we can say that Jana Rae's account balance (J) is equal to her starting amount ($0) plus $25 multiplied by the number of months (t) she has been adding money:
J = 0 + 25t
Now, we can solve the system of equations using the method of substitution or linear combinations.
Using substitution method:
1. Set the two equations equal to each other:
105 + 10t = 0 + 25t
2. Simplify and isolate the variable:
105 = 15t
t = 7
According to the substitution method, after 7 months, Maria and Jana Rae will have the same amount in their accounts.
To find out how much they will have, substitute the value of t back into either equation:
M = 105 + 10 * 7
M = 105 + 70
M = $175
So after 7 months, Maria and Jana Rae will both have $175 in their accounts.
2. To solve the second problem using substitution or linear combinations, we need to create a system of equations that represents the situation of Nate and Gina's video rentals.
Let's use the variables "B" to represent the basic fee and "P" to represent the per-day late fee that the store charges.
For Nate's situation, the total amount paid for his video rental can be expressed as the sum of the basic fee (B) and the per-day late fee (P) multiplied by the number of days (t) he was late:
B + P * t = 5.50
For Gina's situation, the total amount paid for her video rental can be expressed as the sum of the basic fee (B) and the per-day late fee (P) multiplied by the number of days (t) she was late:
B + P * t = 10.00
Now, we can solve the system of equations using the method of substitution or linear combinations.
Using substitution method:
1. Set the two equations equal to each other:
B + P * t = 5.50
B + P * t = 10.00
2. From here, we can see that there is no unique solution to this system of equations. Both equations are not parallel nor coincident, meaning that there exists no unique values for B and P that satisfy both equations simultaneously.
This situation suggests that there may be an error in the problem statement, as it is not possible to determine the basic fee and per-day late fee with the provided information.
Therefore, it is not possible to determine the basic fee and per-day late fee that the store charges in this problem.
just write each sentence in symbols, then solve the equations.
#1
Maria has 105+10m
Jana has 0 + 25m
So, we want
105+10m = 0 + 25m
Now just solve for m and then the amounts
#2
There's a rental rate of r, and a late fee of f, so
Nate: r + 1f = 5.50
Gina: r + 4f = 10.00
Now just solve for r and f
Now just