Question
A bag contains 18 coins consisting of quarters and dimes. The total value of the coins is $2.85. Which system of equations can be used to determine the number of quarters, q, and the number of dimes, d, in the bag?
Question 3 options:
0.25q + 0.10d = 18
q + d = 2.85
0.10q + 0.25d = 2.85
q + d = 18
0.25q + 0.10d = 2.85
q + d = 18
0.10q + 0.25d = 18
q + d = 2.85
Question 3 options:
0.25q + 0.10d = 18
q + d = 2.85
0.10q + 0.25d = 2.85
q + d = 18
0.25q + 0.10d = 2.85
q + d = 18
0.10q + 0.25d = 18
q + d = 2.85
Answers
GPT-4o mini
To set up the correct system of equations, we need to consider two pieces of information: the total number of coins and the total value of the coins.
1. The total number of coins (quarters and dimes) is 18, which gives us the equation:
\[
q + d = 18
\]
2. The total value of the coins is $2.85. Since quarters are worth $0.25 and dimes are worth $0.10, the value equation can be expressed as:
\[
0.25q + 0.10d = 2.85
\]
Therefore, the correct system of equations that can be used to determine the number of quarters \( q \) and the number of dimes \( d \) in the bag is:
\[
\begin{align*}
q + d &= 18 \\
0.25q + 0.10d &= 2.85
\end{align*}
\]
From the given options, this corresponds to:
\[
\text{0.25q + 0.10d = 2.85} \\
\text{q + d = 18}
\]
Thus, the correct answer is **option 3: 0.25q + 0.10d = 2.85 and q + d = 18**.
1. The total number of coins (quarters and dimes) is 18, which gives us the equation:
\[
q + d = 18
\]
2. The total value of the coins is $2.85. Since quarters are worth $0.25 and dimes are worth $0.10, the value equation can be expressed as:
\[
0.25q + 0.10d = 2.85
\]
Therefore, the correct system of equations that can be used to determine the number of quarters \( q \) and the number of dimes \( d \) in the bag is:
\[
\begin{align*}
q + d &= 18 \\
0.25q + 0.10d &= 2.85
\end{align*}
\]
From the given options, this corresponds to:
\[
\text{0.25q + 0.10d = 2.85} \\
\text{q + d = 18}
\]
Thus, the correct answer is **option 3: 0.25q + 0.10d = 2.85 and q + d = 18**.
g7yohulj,
Two customers went to a post office to buy postcards and large envelopes. Each postcard costs the same amount, and each large envelope costs the same amount.
The first customer paid $12 for 14 postcards and 5 large envelopes.
The second customer paid $24.80 for 10 postcards and 15 large envelopes.
What was the cost in dollars of each large envelope?
Question 4 options:
$1.15
$0.35
$0.63
$1.42
The first customer paid $12 for 14 postcards and 5 large envelopes.
The second customer paid $24.80 for 10 postcards and 15 large envelopes.
What was the cost in dollars of each large envelope?
Question 4 options:
$1.15
$0.35
$0.63
$1.42
GPT-4o mini
Let's set up a system of equations to find the cost of each postcard and each large envelope.
Let:
- \( p \) = cost of each postcard (in dollars)
- \( e \) = cost of each large envelope (in dollars)
From the information given about the first customer, we have the equation:
\[
14p + 5e = 12
\]
For the second customer, we have:
\[
10p + 15e = 24.80
\]
Now we can solve this system of equations.
First, let's simplify the first equation:
\[
14p + 5e = 12 \tag{1}
\]
We can multiply this entire equation by 2 to make it easier to eliminate \( p \):
\[
28p + 10e = 24 \tag{2}
\]
Now, let's look at the second equation and also multiply it by 2 to get:
\[
20p + 30e = 49.60 \tag{3}
\]
Now we have:
\[
\begin{align*}
28p + 10e &= 24 \\
20p + 30e &= 49.60
\end{align*}
\]
Next, we can solve either equation for \( p \). Let's start by isolating \( e \) from Equation (2):
\[
10e = 24 - 28p
\]
\[
e = \frac{24 - 28p}{10} \tag{4}
\]
Now substitute (4) into Equation (3):
\[
20p + 30\left(\frac{24 - 28p}{10}\right) = 49.60
\]
Simplifying:
\[
20p + 3(24 - 28p) = 49.60
\]
\[
20p + 72 - 84p = 49.60
\]
\[
-64p + 72 = 49.60
\]
\[
-64p = 49.60 - 72
\]
\[
-64p = -22.40
\]
\[
p = \frac{22.40}{64} \approx 0.35
\]
Next, plug the value of \( p \) back into Equation (4):
\[
e = \frac{24 - 28(0.35)}{10}
\]
\[
e = \frac{24 - 9.8}{10} \approx \frac{14.2}{10} = 1.42
\]
Thus, the cost of each large envelope is:
\[
\boxed{1.42}
\]
So the correct option is **$1.42**.
Let:
- \( p \) = cost of each postcard (in dollars)
- \( e \) = cost of each large envelope (in dollars)
From the information given about the first customer, we have the equation:
\[
14p + 5e = 12
\]
For the second customer, we have:
\[
10p + 15e = 24.80
\]
Now we can solve this system of equations.
First, let's simplify the first equation:
\[
14p + 5e = 12 \tag{1}
\]
We can multiply this entire equation by 2 to make it easier to eliminate \( p \):
\[
28p + 10e = 24 \tag{2}
\]
Now, let's look at the second equation and also multiply it by 2 to get:
\[
20p + 30e = 49.60 \tag{3}
\]
Now we have:
\[
\begin{align*}
28p + 10e &= 24 \\
20p + 30e &= 49.60
\end{align*}
\]
Next, we can solve either equation for \( p \). Let's start by isolating \( e \) from Equation (2):
\[
10e = 24 - 28p
\]
\[
e = \frac{24 - 28p}{10} \tag{4}
\]
Now substitute (4) into Equation (3):
\[
20p + 30\left(\frac{24 - 28p}{10}\right) = 49.60
\]
Simplifying:
\[
20p + 3(24 - 28p) = 49.60
\]
\[
20p + 72 - 84p = 49.60
\]
\[
-64p + 72 = 49.60
\]
\[
-64p = 49.60 - 72
\]
\[
-64p = -22.40
\]
\[
p = \frac{22.40}{64} \approx 0.35
\]
Next, plug the value of \( p \) back into Equation (4):
\[
e = \frac{24 - 28(0.35)}{10}
\]
\[
e = \frac{24 - 9.8}{10} \approx \frac{14.2}{10} = 1.42
\]
Thus, the cost of each large envelope is:
\[
\boxed{1.42}
\]
So the correct option is **$1.42**.
g7yohulj,
What is the solution to this system of equations?
2x + y = 40
x - 2y = -20
Question 5 options:
There are an infinite number of solutions.
(15, 17.5)
There is no solution.
(12, 16)
2x + y = 40
x - 2y = -20
Question 5 options:
There are an infinite number of solutions.
(15, 17.5)
There is no solution.
(12, 16)
GPT-4o mini
To solve the system of equations
1) \( 2x + y = 40 \)
2) \( x - 2y = -20 \)
we can use either substitution or elimination. Here, I'll use the substitution method.
From the first equation (1), we can express \( y \) in terms of \( x \):
\[
y = 40 - 2x \tag{1}
\]
Next, we can substitute \( y \) from equation (1) into the second equation (2):
\[
x - 2(40 - 2x) = -20
\]
Now simplify this expression:
\[
x - 80 + 4x = -20
\]
Combine like terms:
\[
5x - 80 = -20
\]
Add 80 to both sides:
\[
5x = 60
\]
Now divide by 5:
\[
x = 12
\]
Now that we have \( x \), substitute it back into equation (1) to find \( y \):
\[
y = 40 - 2(12) = 40 - 24 = 16
\]
Thus, the solution to the system of equations is:
\[
(x, y) = (12, 16)
\]
Therefore, the correct option is **(12, 16)**.
1) \( 2x + y = 40 \)
2) \( x - 2y = -20 \)
we can use either substitution or elimination. Here, I'll use the substitution method.
From the first equation (1), we can express \( y \) in terms of \( x \):
\[
y = 40 - 2x \tag{1}
\]
Next, we can substitute \( y \) from equation (1) into the second equation (2):
\[
x - 2(40 - 2x) = -20
\]
Now simplify this expression:
\[
x - 80 + 4x = -20
\]
Combine like terms:
\[
5x - 80 = -20
\]
Add 80 to both sides:
\[
5x = 60
\]
Now divide by 5:
\[
x = 12
\]
Now that we have \( x \), substitute it back into equation (1) to find \( y \):
\[
y = 40 - 2(12) = 40 - 24 = 16
\]
Thus, the solution to the system of equations is:
\[
(x, y) = (12, 16)
\]
Therefore, the correct option is **(12, 16)**.
g7yohulj,
What is the value of x in the solution to this system of equations?
6y + x = -59
x = -2y + 9
Question 6 options:
8.5
-12.5
-17
43
6y + x = -59
x = -2y + 9
Question 6 options:
8.5
-12.5
-17
43
GPT-4o mini
To find the value of \( x \) in the given system of equations:
1) \( 6y + x = -59 \)
2) \( x = -2y + 9 \)
We can substitute the expression for \( x \) from the second equation into the first equation.
Substituting \( x \) from equation (2) into equation (1):
\[
6y + (-2y + 9) = -59
\]
Now simplify:
\[
6y - 2y + 9 = -59
\]
\[
4y + 9 = -59
\]
Next, subtract 9 from both sides:
\[
4y = -59 - 9
\]
\[
4y = -68
\]
Now divide by 4:
\[
y = -\frac{68}{4} = -17
\]
Now that we have the value of \( y \), we can substitute it back into either equation to find \( x \). We'll use equation (2):
\[
x = -2y + 9
\]
Substituting \( y = -17 \):
\[
x = -2(-17) + 9
\]
\[
x = 34 + 9
\]
\[
x = 43
\]
Thus, the value of \( x \) in the solution to this system of equations is:
\[
\boxed{43}
\]
So the correct answer is **43**.
1) \( 6y + x = -59 \)
2) \( x = -2y + 9 \)
We can substitute the expression for \( x \) from the second equation into the first equation.
Substituting \( x \) from equation (2) into equation (1):
\[
6y + (-2y + 9) = -59
\]
Now simplify:
\[
6y - 2y + 9 = -59
\]
\[
4y + 9 = -59
\]
Next, subtract 9 from both sides:
\[
4y = -59 - 9
\]
\[
4y = -68
\]
Now divide by 4:
\[
y = -\frac{68}{4} = -17
\]
Now that we have the value of \( y \), we can substitute it back into either equation to find \( x \). We'll use equation (2):
\[
x = -2y + 9
\]
Substituting \( y = -17 \):
\[
x = -2(-17) + 9
\]
\[
x = 34 + 9
\]
\[
x = 43
\]
Thus, the value of \( x \) in the solution to this system of equations is:
\[
\boxed{43}
\]
So the correct answer is **43**.