Question
(4, -3) is the correct solution to which system?
a. 7x - 9y = 55
3x + 9y = -15
b. 7x + 9y = 55
3x - 9y = -15
c. 7x + 9y = 55
3x + 9y = -15
d. 7x - 9y = 55
3x + 9y = -15
a. 7x - 9y = 55
3x + 9y = -15
b. 7x + 9y = 55
3x - 9y = -15
c. 7x + 9y = 55
3x + 9y = -15
d. 7x - 9y = 55
3x + 9y = -15
Answers
Answer
solve the system.
-5x + 4y = 48
-5x - 8y = 24
a. (-24, -18)
b. (-8, 2)
c. ( 8, 2)
d. (24, -18)
-5x + 4y = 48
-5x - 8y = 24
a. (-24, -18)
b. (-8, 2)
c. ( 8, 2)
d. (24, -18)
Answer
solve the system.
8x - 4y = -28
2x - 6y = 8
a. (7, 1)
b. (-2, 3)
c. (-4, -1)
d. (-5, -3)
8x - 4y = -28
2x - 6y = 8
a. (7, 1)
b. (-2, 3)
c. (-4, -1)
d. (-5, -3)
Answer
For renting tables, Parties 'R Us charges, C, a $60 delivery fee and $4 per table, t. Rob's Rentals charges, C, $40 delivery fee and $6 per table, t. Write the pair of linear equations that model this situation.
a. Parties 'R Us: C = 4t + 40; Rob's Rentals: C = 6t + 60
b. Parties 'R Us: C = 4t + 60; Rob's Rentals: C = 6t + 40
c. Parties 'R Us: C = 60t + 6; Rob's Rentals: C = 40t + 4
d. Parties 'R Us: C = 60t + 4; Rob's Rentals: C = 40t + 6
a. Parties 'R Us: C = 4t + 40; Rob's Rentals: C = 6t + 60
b. Parties 'R Us: C = 4t + 60; Rob's Rentals: C = 6t + 40
c. Parties 'R Us: C = 60t + 6; Rob's Rentals: C = 40t + 4
d. Parties 'R Us: C = 60t + 4; Rob's Rentals: C = 40t + 6
Answer
Lincoln High School Student Council wants to raise $300 to make a donation to a homeless shelter. To help the students earn the money, the principal has agreed to pay $15 for each member who picks up litter outside, p, and $10 for each member who cleans the hallways, c. All 25 members of the student council will be participating. Write the pair of linear equations that model this situation.
a. p + c = 300; 15p + 10c = 25
b. p + c = 25; 15p + 10c = 300
c. p + c = 25; 10p + 15c = 300
d. p + c = 300; 10p + 15c = 25
a. p + c = 300; 15p + 10c = 25
b. p + c = 25; 15p + 10c = 300
c. p + c = 25; 10p + 15c = 300
d. p + c = 300; 10p + 15c = 25
Answer
Two boating charter companies operate out of a port. Maritime Voyage charges $250 for the boat rental and $25 per person. Catamaran Command charges $75 for the boat rental and $50 per person. If a group has 10 customers, which charter should they choose?
a. Maritime Voyage; choose this charter for less than 13 customers.
b. Maritime Voyage; choose this charter for more than 7 customers.
c. Catamaran Command; choose this charter for more than 7 customers.
d. Catamaran Command; choose this charter for less than 13 customers.
a. Maritime Voyage; choose this charter for less than 13 customers.
b. Maritime Voyage; choose this charter for more than 7 customers.
c. Catamaran Command; choose this charter for more than 7 customers.
d. Catamaran Command; choose this charter for less than 13 customers.
Answered by
GPT-4o mini
To determine which system of equations is satisfied by the point (4, -3), we substitute \( x = 4 \) and \( y = -3 \) into each system.
Let's evaluate each option:
**Option a:**
1. \( 7x - 9y = 55 \)
\( 7(4) - 9(-3) = 28 + 27 = 55 \) (True)
2. \( 3x + 9y = -15 \)
\( 3(4) + 9(-3) = 12 - 27 = -15 \) (True)
Both equations are satisfied, so option (a) is a solution.
**Option b:**
1. \( 7x + 9y = 55 \)
\( 7(4) + 9(-3) = 28 - 27 = 1 \) (False)
2. \( 3x - 9y = -15 \)
\( 3(4) - 9(-3) = 12 + 27 = 39 \) (False)
Both equations are not satisfied.
**Option c:**
1. \( 7x + 9y = 55 \)
\( 7(4) + 9(-3) = 28 - 27 = 1 \) (False)
2. \( 3x + 9y = -15 \)
\( 3(4) + 9(-3) = 12 - 27 = -15 \) (True)
One equation is not satisfied.
**Option d:**
1. \( 7x - 9y = 55 \)
\( 7(4) - 9(-3) = 28 + 27 = 55 \) (True)
2. \( 3x + 9y = -15 \)
\( 3(4) + 9(-3) = 12 - 27 = -15 \) (True)
Both equations are satisfied, so option (d) is also a solution.
Thus, (4, -3) is the correct solution for both **Option a** and **Option d**. If you are looking for a single option that is correct based on typical conventions, you would choose:
**Answer: a. 7x - 9y = 55; 3x + 9y = -15**.
Let's evaluate each option:
**Option a:**
1. \( 7x - 9y = 55 \)
\( 7(4) - 9(-3) = 28 + 27 = 55 \) (True)
2. \( 3x + 9y = -15 \)
\( 3(4) + 9(-3) = 12 - 27 = -15 \) (True)
Both equations are satisfied, so option (a) is a solution.
**Option b:**
1. \( 7x + 9y = 55 \)
\( 7(4) + 9(-3) = 28 - 27 = 1 \) (False)
2. \( 3x - 9y = -15 \)
\( 3(4) - 9(-3) = 12 + 27 = 39 \) (False)
Both equations are not satisfied.
**Option c:**
1. \( 7x + 9y = 55 \)
\( 7(4) + 9(-3) = 28 - 27 = 1 \) (False)
2. \( 3x + 9y = -15 \)
\( 3(4) + 9(-3) = 12 - 27 = -15 \) (True)
One equation is not satisfied.
**Option d:**
1. \( 7x - 9y = 55 \)
\( 7(4) - 9(-3) = 28 + 27 = 55 \) (True)
2. \( 3x + 9y = -15 \)
\( 3(4) + 9(-3) = 12 - 27 = -15 \) (True)
Both equations are satisfied, so option (d) is also a solution.
Thus, (4, -3) is the correct solution for both **Option a** and **Option d**. If you are looking for a single option that is correct based on typical conventions, you would choose:
**Answer: a. 7x - 9y = 55; 3x + 9y = -15**.
Answered by
GPT-4o mini
To solve the system of equations given:
1. \(-5x + 4y = 48\)
2. \(-5x - 8y = 24\)
We can use the method of elimination or substitution. Let's use elimination here.
First, we can align the equations:
1. \(-5x + 4y = 48\) (Equation 1)
2. \(-5x - 8y = 24\) (Equation 2)
We can eliminate \(x\) by subtracting Equation 2 from Equation 1:
\[
(-5x + 4y) - (-5x - 8y) = 48 - 24
\]
This simplifies to:
\[
4y + 8y = 24
\]
Combining the terms gives:
\[
12y = 24
\]
Now, solve for \(y\):
\[
y = \frac{24}{12} = 2
\]
Now that we have \(y\), we can substitute it back into one of the original equations to find \(x\). We'll use Equation 1:
\[
-5x + 4(2) = 48
\]
This simplifies to:
\[
-5x + 8 = 48
\]
Now, isolate \(x\):
\[
-5x = 48 - 8
\]
\[
-5x = 40
\]
\[
x = \frac{40}{-5} = -8
\]
So, the solution to the system is:
\[
(x, y) = (-8, 2)
\]
Thus, the answer is:
**b. (-8, 2)**.
1. \(-5x + 4y = 48\)
2. \(-5x - 8y = 24\)
We can use the method of elimination or substitution. Let's use elimination here.
First, we can align the equations:
1. \(-5x + 4y = 48\) (Equation 1)
2. \(-5x - 8y = 24\) (Equation 2)
We can eliminate \(x\) by subtracting Equation 2 from Equation 1:
\[
(-5x + 4y) - (-5x - 8y) = 48 - 24
\]
This simplifies to:
\[
4y + 8y = 24
\]
Combining the terms gives:
\[
12y = 24
\]
Now, solve for \(y\):
\[
y = \frac{24}{12} = 2
\]
Now that we have \(y\), we can substitute it back into one of the original equations to find \(x\). We'll use Equation 1:
\[
-5x + 4(2) = 48
\]
This simplifies to:
\[
-5x + 8 = 48
\]
Now, isolate \(x\):
\[
-5x = 48 - 8
\]
\[
-5x = 40
\]
\[
x = \frac{40}{-5} = -8
\]
So, the solution to the system is:
\[
(x, y) = (-8, 2)
\]
Thus, the answer is:
**b. (-8, 2)**.
Answered by
GPT-4o mini
To solve the system of equations given:
1. \(8x - 4y = -28\)
2. \(2x - 6y = 8\)
We can use either the substitution method or the elimination method. I'll use the elimination method here.
### Step 1: Simplifying the First Equation
We can simplify the first equation by dividing everything by 4:
\[
2x - y = -7 \quad \text{(Equation 1)}
\]
### Step 2: Keeping the Second Equation the Same
The second equation remains:
\[
2x - 6y = 8 \quad \text{(Equation 2)}
\]
### Step 3: Elimination of \(x\)
We can eliminate \(x\) by subtracting Equation 1 from Equation 2.
First, manipulation should look like this:
1. Let’s align the equations:
\[
2x - y = -7
\]
\[
2x - 6y = 8
\]
2. Subtract Equation 1 from Equation 2:
\[
(2x - 6y) - (2x - y) = 8 - (-7)
\]
This simplifies to:
\[
-6y + y = 8 + 7
\]
Combining the terms gives:
\[
-5y = 15
\]
### Step 4: Solve for \(y\)
\[
y = \frac{15}{-5} = -3
\]
### Step 5: Substitute \(y\) Back to Find \(x\)
Now that we have \(y = -3\), substitute it back into Equation 1 to find \(x\):
\[
2x - (-3) = -7
\]
This simplifies to:
\[
2x + 3 = -7
\]
Now isolate \(x\):
\[
2x = -7 - 3
\]
\[
2x = -10
\]
\[
x = \frac{-10}{2} = -5
\]
### Final Solution
So the solution to the system is:
\[
(x, y) = (-5, -3)
\]
Thus, the correct answer is:
**d. (-5, -3)**.
1. \(8x - 4y = -28\)
2. \(2x - 6y = 8\)
We can use either the substitution method or the elimination method. I'll use the elimination method here.
### Step 1: Simplifying the First Equation
We can simplify the first equation by dividing everything by 4:
\[
2x - y = -7 \quad \text{(Equation 1)}
\]
### Step 2: Keeping the Second Equation the Same
The second equation remains:
\[
2x - 6y = 8 \quad \text{(Equation 2)}
\]
### Step 3: Elimination of \(x\)
We can eliminate \(x\) by subtracting Equation 1 from Equation 2.
First, manipulation should look like this:
1. Let’s align the equations:
\[
2x - y = -7
\]
\[
2x - 6y = 8
\]
2. Subtract Equation 1 from Equation 2:
\[
(2x - 6y) - (2x - y) = 8 - (-7)
\]
This simplifies to:
\[
-6y + y = 8 + 7
\]
Combining the terms gives:
\[
-5y = 15
\]
### Step 4: Solve for \(y\)
\[
y = \frac{15}{-5} = -3
\]
### Step 5: Substitute \(y\) Back to Find \(x\)
Now that we have \(y = -3\), substitute it back into Equation 1 to find \(x\):
\[
2x - (-3) = -7
\]
This simplifies to:
\[
2x + 3 = -7
\]
Now isolate \(x\):
\[
2x = -7 - 3
\]
\[
2x = -10
\]
\[
x = \frac{-10}{2} = -5
\]
### Final Solution
So the solution to the system is:
\[
(x, y) = (-5, -3)
\]
Thus, the correct answer is:
**d. (-5, -3)**.
Answered by
GPT-4o mini
To write the pair of linear equations that model the renting tables situation for Parties 'R Us and Rob's Rentals, we will define the cost \( C \) based on the number of tables \( t \).
### For Parties 'R Us:
- They charge a **$60 delivery fee** and **$4 per table**.
Thus, the equation becomes:
\[
C = 4t + 60
\]
### For Rob's Rentals:
- They charge a **$40 delivery fee** and **$6 per table**.
Thus, the equation becomes:
\[
C = 6t + 40
\]
### Therefore, the pair of linear equations is:
1. Parties 'R Us: \( C = 4t + 60 \)
2. Rob's Rentals: \( C = 6t + 40 \)
Now, comparing this with the options provided:
**Answer: b. Parties 'R Us: C = 4t + 60; Rob's Rentals: C = 6t + 40**.
### For Parties 'R Us:
- They charge a **$60 delivery fee** and **$4 per table**.
Thus, the equation becomes:
\[
C = 4t + 60
\]
### For Rob's Rentals:
- They charge a **$40 delivery fee** and **$6 per table**.
Thus, the equation becomes:
\[
C = 6t + 40
\]
### Therefore, the pair of linear equations is:
1. Parties 'R Us: \( C = 4t + 60 \)
2. Rob's Rentals: \( C = 6t + 40 \)
Now, comparing this with the options provided:
**Answer: b. Parties 'R Us: C = 4t + 60; Rob's Rentals: C = 6t + 40**.
Answered by
GPT-4o mini
To write the pair of linear equations that model the situation, we need to define two variables:
- Let **\( p \)** be the number of members picking up litter.
- Let **\( c \)** be the number of members cleaning the hallways.
### Given Information:
1. The total number of members participating is **25**.
2. The total amount of money to raise is **$300**.
3. Each member who picks up litter earns **$15**.
4. Each member who cleans the hallways earns **$10**.
### First Equation:
The first equation represents the total number of members participating:
\[
p + c = 25
\]
### Second Equation:
The second equation represents the total amount of money raised by the students:
\[
15p + 10c = 300
\]
### Therefore, the pair of linear equations is:
1. \( p + c = 25 \)
2. \( 15p + 10c = 300 \)
Now, let's compare this with the options provided:
**Answer: b. \( p + c = 25; \, 15p + 10c = 300 \)**.
- Let **\( p \)** be the number of members picking up litter.
- Let **\( c \)** be the number of members cleaning the hallways.
### Given Information:
1. The total number of members participating is **25**.
2. The total amount of money to raise is **$300**.
3. Each member who picks up litter earns **$15**.
4. Each member who cleans the hallways earns **$10**.
### First Equation:
The first equation represents the total number of members participating:
\[
p + c = 25
\]
### Second Equation:
The second equation represents the total amount of money raised by the students:
\[
15p + 10c = 300
\]
### Therefore, the pair of linear equations is:
1. \( p + c = 25 \)
2. \( 15p + 10c = 300 \)
Now, let's compare this with the options provided:
**Answer: b. \( p + c = 25; \, 15p + 10c = 300 \)**.
Answered by
GPT-4o mini
To determine which charter company the group should choose, we need to calculate the total cost for both Maritime Voyage and Catamaran Command when there are 10 customers.
### For Maritime Voyage:
- Rental fee: $250
- Cost per person: $25
- Total number of customers: 10
The total cost for Maritime Voyage is calculated as follows:
\[
\text{Total Cost} = \text{Rental Fee} + (\text{Cost per Person} \times \text{Number of Customers})
\]
\[
\text{Total Cost} = 250 + (25 \times 10) = 250 + 250 = 500
\]
### For Catamaran Command:
- Rental fee: $75
- Cost per person: $50
- Total number of customers: 10
The total cost for Catamaran Command is calculated as:
\[
\text{Total Cost} = \text{Rental Fee} + (\text{Cost per Person} \times \text{Number of Customers})
\]
\[
\text{Total Cost} = 75 + (50 \times 10) = 75 + 500 = 575
\]
### Conclusion:
- Total cost for Maritime Voyage: **$500**
- Total cost for Catamaran Command: **$575**
Since **$500 < $575**, the group should choose **Maritime Voyage**.
### Analyzing the Options:
Now let's look at the options to determine which one is the correct choice based on this analysis.
**Answer:**
**a. Maritime Voyage; choose this charter for less than 13 customers.**
This option is correct because Maritime Voyage is cheaper for the group of 10 customers, and increasing the number of customers beyond 13 would continue favoring Maritime Voyage due to its lower base fee compared to Catamaran Command.
### For Maritime Voyage:
- Rental fee: $250
- Cost per person: $25
- Total number of customers: 10
The total cost for Maritime Voyage is calculated as follows:
\[
\text{Total Cost} = \text{Rental Fee} + (\text{Cost per Person} \times \text{Number of Customers})
\]
\[
\text{Total Cost} = 250 + (25 \times 10) = 250 + 250 = 500
\]
### For Catamaran Command:
- Rental fee: $75
- Cost per person: $50
- Total number of customers: 10
The total cost for Catamaran Command is calculated as:
\[
\text{Total Cost} = \text{Rental Fee} + (\text{Cost per Person} \times \text{Number of Customers})
\]
\[
\text{Total Cost} = 75 + (50 \times 10) = 75 + 500 = 575
\]
### Conclusion:
- Total cost for Maritime Voyage: **$500**
- Total cost for Catamaran Command: **$575**
Since **$500 < $575**, the group should choose **Maritime Voyage**.
### Analyzing the Options:
Now let's look at the options to determine which one is the correct choice based on this analysis.
**Answer:**
**a. Maritime Voyage; choose this charter for less than 13 customers.**
This option is correct because Maritime Voyage is cheaper for the group of 10 customers, and increasing the number of customers beyond 13 would continue favoring Maritime Voyage due to its lower base fee compared to Catamaran Command.
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