Question
Write two equations:
#1 The difference between the two numbers is -10.
#2 The sum of the two numbers is -4.
Let x = _____________
Let y =____________
#1 The difference between the two numbers is -10.
#2 The sum of the two numbers is -4.
Let x = _____________
Let y =____________
Answers
GPT-4o mini
We can express the two equations based on the statements provided.
1. The difference between the two numbers is -10:
\[
x - y = -10
\]
2. The sum of the two numbers is -4:
\[
x + y = -4
\]
Now to write it in the requested format:
Let \( x = \) (the first number)
Let \( y = \) (the second number)
So, the complete set becomes:
Let \( x = \) the first number
Let \( y = \) the second number
The equations are:
1. \( x - y = -10 \)
2. \( x + y = -4 \)
1. The difference between the two numbers is -10:
\[
x - y = -10
\]
2. The sum of the two numbers is -4:
\[
x + y = -4
\]
Now to write it in the requested format:
Let \( x = \) (the first number)
Let \( y = \) (the second number)
So, the complete set becomes:
Let \( x = \) the first number
Let \( y = \) the second number
The equations are:
1. \( x - y = -10 \)
2. \( x + y = -4 \)
BTS ARMY
Write two equations:
#1 Jack and his sister, Malonie, are 4 years apart in age.
#2 The sum of their ages is 28.
What are their ages?
Let ___ = _____________
Let ___=______________
#1 Jack and his sister, Malonie, are 4 years apart in age.
#2 The sum of their ages is 28.
What are their ages?
Let ___ = _____________
Let ___=______________
GPT-4o mini
Let's define the variables for their ages:
Let \( j = \) Jack's age
Let \( m = \) Malonie's age
Now we can write the equations based on the provided statements:
1. Jack and his sister, Malonie, are 4 years apart in age:
\[
j - m = 4 \quad \text{(this means Jack is older)}
\]
Alternatively, it can be expressed as:
\[
m - j = 4 \quad \text{(if Malonie is older)}
\]
2. The sum of their ages is 28:
\[
j + m = 28
\]
To summarize:
Let \( j = \) Jack's age
Let \( m = \) Malonie's age
The equations are:
1. \( j - m = 4 \) (assuming Jack is older)
2. \( j + m = 28 \)
Now, to solve for their ages, we can use the two equations:
From the first equation, we can express \( j \) in terms of \( m \):
\[
j = m + 4
\]
Now substitute \( j \) in the second equation:
\[
(m + 4) + m = 28
\]
Combine like terms:
\[
2m + 4 = 28
\]
Subtract 4 from both sides:
\[
2m = 24
\]
Divide by 2:
\[
m = 12
\]
Now substitute \( m \) back to find \( j \):
\[
j = m + 4 = 12 + 4 = 16
\]
Thus, Jack's age is 16 and Malonie's age is 12.
Final answer:
Jack's age: 16
Malonie's age: 12
Let \( j = \) Jack's age
Let \( m = \) Malonie's age
Now we can write the equations based on the provided statements:
1. Jack and his sister, Malonie, are 4 years apart in age:
\[
j - m = 4 \quad \text{(this means Jack is older)}
\]
Alternatively, it can be expressed as:
\[
m - j = 4 \quad \text{(if Malonie is older)}
\]
2. The sum of their ages is 28:
\[
j + m = 28
\]
To summarize:
Let \( j = \) Jack's age
Let \( m = \) Malonie's age
The equations are:
1. \( j - m = 4 \) (assuming Jack is older)
2. \( j + m = 28 \)
Now, to solve for their ages, we can use the two equations:
From the first equation, we can express \( j \) in terms of \( m \):
\[
j = m + 4
\]
Now substitute \( j \) in the second equation:
\[
(m + 4) + m = 28
\]
Combine like terms:
\[
2m + 4 = 28
\]
Subtract 4 from both sides:
\[
2m = 24
\]
Divide by 2:
\[
m = 12
\]
Now substitute \( m \) back to find \( j \):
\[
j = m + 4 = 12 + 4 = 16
\]
Thus, Jack's age is 16 and Malonie's age is 12.
Final answer:
Jack's age: 16
Malonie's age: 12
BTS ARMY
Write two equations:
#1 Two of Julie’s textbooks are a total of $65.
#2 The difference in price between the two books is $9.
What is the cost of each book?
Let ___ = _____________
Let ___=______________
#1 Two of Julie’s textbooks are a total of $65.
#2 The difference in price between the two books is $9.
What is the cost of each book?
Let ___ = _____________
Let ___=______________
GPT-4o mini
Let's define the variables for the prices of the textbooks:
Let \( x = \) the cost of the first textbook
Let \( y = \) the cost of the second textbook
Now we can write the equations based on the provided statements:
1. Two of Julie’s textbooks are a total of $65:
\[
x + y = 65
\]
2. The difference in price between the two books is $9:
\[
x - y = 9 \quad \text{(assuming the first book is more expensive)}
\]
Now we can summarize:
Let \( x = \) the cost of the first textbook
Let \( y = \) the cost of the second textbook
The equations are:
1. \( x + y = 65 \)
2. \( x - y = 9 \)
### Solving the equations
We can use the first equation to express \( y \) in terms of \( x \):
\[
y = 65 - x
\]
Now, substitute \( y \) in the second equation:
\[
x - (65 - x) = 9
\]
Simplify:
\[
x - 65 + x = 9
\]
Combine like terms:
\[
2x - 65 = 9
\]
Add 65 to both sides:
\[
2x = 74
\]
Divide by 2:
\[
x = 37
\]
Now substitute \( x \) back to find \( y \):
\[
y = 65 - x = 65 - 37 = 28
\]
### Final Answer:
The cost of the first textbook is $37, and the cost of the second textbook is $28.
So,
Let \( x = 37 \) (cost of the first textbook)
Let \( y = 28 \) (cost of the second textbook)
Let \( x = \) the cost of the first textbook
Let \( y = \) the cost of the second textbook
Now we can write the equations based on the provided statements:
1. Two of Julie’s textbooks are a total of $65:
\[
x + y = 65
\]
2. The difference in price between the two books is $9:
\[
x - y = 9 \quad \text{(assuming the first book is more expensive)}
\]
Now we can summarize:
Let \( x = \) the cost of the first textbook
Let \( y = \) the cost of the second textbook
The equations are:
1. \( x + y = 65 \)
2. \( x - y = 9 \)
### Solving the equations
We can use the first equation to express \( y \) in terms of \( x \):
\[
y = 65 - x
\]
Now, substitute \( y \) in the second equation:
\[
x - (65 - x) = 9
\]
Simplify:
\[
x - 65 + x = 9
\]
Combine like terms:
\[
2x - 65 = 9
\]
Add 65 to both sides:
\[
2x = 74
\]
Divide by 2:
\[
x = 37
\]
Now substitute \( x \) back to find \( y \):
\[
y = 65 - x = 65 - 37 = 28
\]
### Final Answer:
The cost of the first textbook is $37, and the cost of the second textbook is $28.
So,
Let \( x = 37 \) (cost of the first textbook)
Let \( y = 28 \) (cost of the second textbook)
BTS ARMY
Write a system of equations where the product of two numbers is 10 and the first number is 3 more than the second.
Can you determine the solution? Guess and check.
Can you determine the solution? Guess and check.
GPT-4o mini
Let's define the variables:
Let \( x = \) the first number
Let \( y = \) the second number
Now we can write the system of equations based on the given information:
1. The product of two numbers is 10:
\[
xy = 10
\]
2. The first number is 3 more than the second:
\[
x = y + 3
\]
### System of Equations:
1. \( xy = 10 \)
2. \( x = y + 3 \)
### Substitute and Solve using Guess and Check
We can substitute the second equation into the first equation:
Substituting \( x \) from the second equation into the first:
\[
(y + 3)y = 10
\]
Expanding this gives:
\[
y^2 + 3y - 10 = 0
\]
To solve this quadratic equation using guess and check, we can look for two numbers that multiply to \(-10\) and add up to \(3\).
Let's try some values:
1. If \( y = 2 \):
\[
x = 2 + 3 = 5
\]
Calculate the product:
\[
5 \cdot 2 = 10 \quad (\text{This works!})
\]
2. If \( y = -5 \) (checking for completeness):
\[
x = -5 + 3 = -2
\]
Calculate the product:
\[
-2 \cdot -5 = 10 \quad (\text{This also works!})
\]
### Solutions:
1. The numbers can be \( (5, 2) \) or \( (-2, -5) \).
Thus, the solutions to the system of equations are:
- **First solution**: \( x = 5 \) and \( y = 2 \)
- **Second solution**: \( x = -2 \) and \( y = -5 \)
Let \( x = \) the first number
Let \( y = \) the second number
Now we can write the system of equations based on the given information:
1. The product of two numbers is 10:
\[
xy = 10
\]
2. The first number is 3 more than the second:
\[
x = y + 3
\]
### System of Equations:
1. \( xy = 10 \)
2. \( x = y + 3 \)
### Substitute and Solve using Guess and Check
We can substitute the second equation into the first equation:
Substituting \( x \) from the second equation into the first:
\[
(y + 3)y = 10
\]
Expanding this gives:
\[
y^2 + 3y - 10 = 0
\]
To solve this quadratic equation using guess and check, we can look for two numbers that multiply to \(-10\) and add up to \(3\).
Let's try some values:
1. If \( y = 2 \):
\[
x = 2 + 3 = 5
\]
Calculate the product:
\[
5 \cdot 2 = 10 \quad (\text{This works!})
\]
2. If \( y = -5 \) (checking for completeness):
\[
x = -5 + 3 = -2
\]
Calculate the product:
\[
-2 \cdot -5 = 10 \quad (\text{This also works!})
\]
### Solutions:
1. The numbers can be \( (5, 2) \) or \( (-2, -5) \).
Thus, the solutions to the system of equations are:
- **First solution**: \( x = 5 \) and \( y = 2 \)
- **Second solution**: \( x = -2 \) and \( y = -5 \)
BTS ARMY
Tell whether the ordered pair is a solution of the equation. Just substitute the given x and y to see if the equation “works”. Write “solution” if it works and “not solution” if it doesn’t.
y=4x + 2; (2,10)
2x + y = 5; (7, 5)
y=4x + 2; (2,10)
2x + y = 5; (7, 5)
GPT-4o mini
To determine if the ordered pairs are solutions for the given equations, we can substitute the values of \( x \) and \( y \) into each equation and see if both sides are equal.
### For the first equation:
**Equation:** \( y = 4x + 2 \)
**Ordered Pair:** (2, 10)
Substituting \( x = 2 \) and \( y = 10 \) into the equation:
\[
10 = 4(2) + 2
\]
\[
10 = 8 + 2
\]
\[
10 = 10 \quad (\text{True})
\]
Since both sides are equal, the ordered pair (2, 10) **is a solution**.
### For the second equation:
**Equation:** \( 2x + y = 5 \)
**Ordered Pair:** (7, 5)
Substituting \( x = 7 \) and \( y = 5 \) into the equation:
\[
2(7) + 5 = 5
\]
\[
14 + 5 = 5
\]
\[
19 = 5 \quad (\text{False})
\]
Since both sides are not equal, the ordered pair (7, 5) **is not a solution**.
### Summary:
1. For \( y = 4x + 2; (2, 10) \) → **solution**
2. For \( 2x + y = 5; (7, 5) \) → **not solution**
### For the first equation:
**Equation:** \( y = 4x + 2 \)
**Ordered Pair:** (2, 10)
Substituting \( x = 2 \) and \( y = 10 \) into the equation:
\[
10 = 4(2) + 2
\]
\[
10 = 8 + 2
\]
\[
10 = 10 \quad (\text{True})
\]
Since both sides are equal, the ordered pair (2, 10) **is a solution**.
### For the second equation:
**Equation:** \( 2x + y = 5 \)
**Ordered Pair:** (7, 5)
Substituting \( x = 7 \) and \( y = 5 \) into the equation:
\[
2(7) + 5 = 5
\]
\[
14 + 5 = 5
\]
\[
19 = 5 \quad (\text{False})
\]
Since both sides are not equal, the ordered pair (7, 5) **is not a solution**.
### Summary:
1. For \( y = 4x + 2; (2, 10) \) → **solution**
2. For \( 2x + y = 5; (7, 5) \) → **not solution**
BTS ARMY
Set up a table of values:
Y(1) = 1/2x + 5
Y(2) = -2x -1
x
y(1)=
1/2x+5
y(2)=
-2x - 1
(x,y(1))
(x, y(2))
8
4
0
-2
Y(1) = 1/2x + 5
Y(2) = -2x -1
x
y(1)=
1/2x+5
y(2)=
-2x - 1
(x,y(1))
(x, y(2))
8
4
0
-2
GPT-4o mini
To create a table of values for the functions \( Y(1) = \frac{1}{2}x + 5 \) and \( Y(2) = -2x - 1 \), we will substitute the specified \( x \) values (8, 4, 0, and -2) into both equations and calculate the corresponding \( y \) values.
### Let's compute the values:
1. **For \( x = 8 \)**:
- \( Y(1) = \frac{1}{2}(8) + 5 = 4 + 5 = 9 \)
- \( Y(2) = -2(8) - 1 = -16 - 1 = -17 \)
2. **For \( x = 4 \)**:
- \( Y(1) = \frac{1}{2}(4) + 5 = 2 + 5 = 7 \)
- \( Y(2) = -2(4) - 1 = -8 - 1 = -9 \)
3. **For \( x = 0 \)**:
- \( Y(1) = \frac{1}{2}(0) + 5 = 0 + 5 = 5 \)
- \( Y(2) = -2(0) - 1 = 0 - 1 = -1 \)
4. **For \( x = -2 \)**:
- \( Y(1) = \frac{1}{2}(-2) + 5 = -1 + 5 = 4 \)
- \( Y(2) = -2(-2) - 1 = 4 - 1 = 3 \)
### Summarizing in a table:
\[
\begin{array}{|c|c|c|}
\hline
x & y(1) = \frac{1}{2}x + 5 & y(2) = -2x - 1 \\
\hline
8 & 9 & -17 \\
4 & 7 & -9 \\
0 & 5 & -1 \\
-2 & 4 & 3 \\
\hline
\end{array}
\]
### Final Table of Values:
\[
\begin{array}{|c|c|c|}
\hline
x & y(1) & y(2) \\
\hline
8 & 9 & -17 \\
4 & 7 & -9 \\
0 & 5 & -1 \\
-2 & 4 & 3 \\
\hline
\end{array}
\]
This table summarizes the calculated \( y \) values for each corresponding \( x \).
### Let's compute the values:
1. **For \( x = 8 \)**:
- \( Y(1) = \frac{1}{2}(8) + 5 = 4 + 5 = 9 \)
- \( Y(2) = -2(8) - 1 = -16 - 1 = -17 \)
2. **For \( x = 4 \)**:
- \( Y(1) = \frac{1}{2}(4) + 5 = 2 + 5 = 7 \)
- \( Y(2) = -2(4) - 1 = -8 - 1 = -9 \)
3. **For \( x = 0 \)**:
- \( Y(1) = \frac{1}{2}(0) + 5 = 0 + 5 = 5 \)
- \( Y(2) = -2(0) - 1 = 0 - 1 = -1 \)
4. **For \( x = -2 \)**:
- \( Y(1) = \frac{1}{2}(-2) + 5 = -1 + 5 = 4 \)
- \( Y(2) = -2(-2) - 1 = 4 - 1 = 3 \)
### Summarizing in a table:
\[
\begin{array}{|c|c|c|}
\hline
x & y(1) = \frac{1}{2}x + 5 & y(2) = -2x - 1 \\
\hline
8 & 9 & -17 \\
4 & 7 & -9 \\
0 & 5 & -1 \\
-2 & 4 & 3 \\
\hline
\end{array}
\]
### Final Table of Values:
\[
\begin{array}{|c|c|c|}
\hline
x & y(1) & y(2) \\
\hline
8 & 9 & -17 \\
4 & 7 & -9 \\
0 & 5 & -1 \\
-2 & 4 & 3 \\
\hline
\end{array}
\]
This table summarizes the calculated \( y \) values for each corresponding \( x \).