Question
Q 1. Is the equation true, false, or open? 3x + 1 = 22
Answers
GPT 3.5
The equation is true when x equals 7.
Which value is a solution of the equation 4 - 2x = 8?(1 point) Responses
GPT 3.5
To find the value that is a solution of the equation 4 - 2x = 8, we can solve for x.
First, subtract 4 from both sides of the equation:
4 - 2x - 4 = 8 - 4
This simplifies to:
-2x = 4
Next, divide both sides of the equation by -2:
-2x/-2 = 4/-2
This simplifies to:
x = -2
Therefore, the value -2 is a solution of the equation 4 - 2x = 8.
First, subtract 4 from both sides of the equation:
4 - 2x - 4 = 8 - 4
This simplifies to:
-2x = 4
Next, divide both sides of the equation by -2:
-2x/-2 = 4/-2
This simplifies to:
x = -2
Therefore, the value -2 is a solution of the equation 4 - 2x = 8.
Q 3. Solve the following equation: m - 8 = 14(1 point)
GPT 3.5
To solve the equation m - 8 = 14, you can isolate the variable m by adding 8 to both sides of the equation:
m - 8 + 8 = 14 + 8
This simplifies to:
m = 22
Therefore, the solution to the equation m - 8 = 14 is m = 22.
m - 8 + 8 = 14 + 8
This simplifies to:
m = 22
Therefore, the solution to the equation m - 8 = 14 is m = 22.
Q 4. Which ordered pair is a solution of y = x - 5?(1 point)
GPT 3.5
To find which ordered pair is a solution of y = x - 5, we need to substitute different values for x and see if the resulting y values satisfy the equation.
Let's try a few different values of x:
For x = 0:
y = 0 - 5
y = -5
So the ordered pair (0, -5) satisfies the equation.
For x = 1:
y = 1 - 5
y = -4
So the ordered pair (1, -4) satisfies the equation.
For x = 2:
y = 2 - 5
y = -3
So the ordered pair (2, -3) satisfies the equation.
From our calculations, the ordered pairs (0, -5), (1, -4), and (2, -3) are all solutions of the equation y = x - 5.
Let's try a few different values of x:
For x = 0:
y = 0 - 5
y = -5
So the ordered pair (0, -5) satisfies the equation.
For x = 1:
y = 1 - 5
y = -4
So the ordered pair (1, -4) satisfies the equation.
For x = 2:
y = 2 - 5
y = -3
So the ordered pair (2, -3) satisfies the equation.
From our calculations, the ordered pairs (0, -5), (1, -4), and (2, -3) are all solutions of the equation y = x - 5.
Q. 5. Solve the following equation: 10 + x3=1 (1 point)
GPT 3.5
To solve the equation 10 + x^3 = 1, we need to isolate the variable x.
First, subtract 10 from both sides of the equation:
10 + x^3 - 10 = 1 - 10
This simplifies to:
x^3 = -9
Next, we can take the cube root of both sides of the equation to solve for x:
∛(x^3) = ∛(-9)
This simplifies to:
x = -∛9
Therefore, the solution to the equation 10 + x^3 = 1 is x = -∛9.
First, subtract 10 from both sides of the equation:
10 + x^3 - 10 = 1 - 10
This simplifies to:
x^3 = -9
Next, we can take the cube root of both sides of the equation to solve for x:
∛(x^3) = ∛(-9)
This simplifies to:
x = -∛9
Therefore, the solution to the equation 10 + x^3 = 1 is x = -∛9.
Q. 6. Solve the following equation: 7 (3x - 4) = 49(1
GPT 3.5
To solve the equation 7(3x - 4) = 49, we will follow the steps:
First, distribute the 7 to the terms inside the parentheses:
21x - 28 = 49
Next, add 28 to both sides of the equation to isolate the 21x term:
21x - 28 + 28 = 49 + 28
This simplifies to:
21x = 77
Finally, divide both sides of the equation by 21 to solve for x:
21x/21 = 77/21
This simplifies to:
x = 3.6667 (rounded to four decimal places)
Therefore, the solution to the equation 7(3x - 4) = 49 is x ≈ 3.6667.
First, distribute the 7 to the terms inside the parentheses:
21x - 28 = 49
Next, add 28 to both sides of the equation to isolate the 21x term:
21x - 28 + 28 = 49 + 28
This simplifies to:
21x = 77
Finally, divide both sides of the equation by 21 to solve for x:
21x/21 = 77/21
This simplifies to:
x = 3.6667 (rounded to four decimal places)
Therefore, the solution to the equation 7(3x - 4) = 49 is x ≈ 3.6667.
Q. 7. Solve the following equation: 1.3 + 0.5x = -3.41(1 point)
GPT 3.5
To solve the equation 1.3 + 0.5x = -3.41, we will follow the steps:
First, subtract 1.3 from both sides of the equation to isolate the 0.5x term:
1.3 + 0.5x - 1.3 = -3.41 - 1.3
This simplifies to:
0.5x = -4.71
Next, divide both sides of the equation by 0.5 to solve for x:
0.5x / 0.5 = -4.71 / 0.5
This simplifies to:
x = -9.42
Therefore, the solution to the equation 1.3 + 0.5x = -3.41 is x = -9.42.
First, subtract 1.3 from both sides of the equation to isolate the 0.5x term:
1.3 + 0.5x - 1.3 = -3.41 - 1.3
This simplifies to:
0.5x = -4.71
Next, divide both sides of the equation by 0.5 to solve for x:
0.5x / 0.5 = -4.71 / 0.5
This simplifies to:
x = -9.42
Therefore, the solution to the equation 1.3 + 0.5x = -3.41 is x = -9.42.
Q. 8. Solve the following equation: (413) x = 9 (1 point) Responses
GPT 3.5
To solve the equation (413)x = 9, we will follow the steps:
Since there is no specific value provided for x, we need to solve for it.
To do that, we need to isolate x by dividing both sides of the equation by 413:
(413)x / 413 = 9 / 413
This simplifies to:
x = 0.0218 (rounded to four decimal places)
Therefore, the solution to the equation (413)x = 9 is x ≈ 0.0218.
Since there is no specific value provided for x, we need to solve for it.
To do that, we need to isolate x by dividing both sides of the equation by 413:
(413)x / 413 = 9 / 413
This simplifies to:
x = 0.0218 (rounded to four decimal places)
Therefore, the solution to the equation (413)x = 9 is x ≈ 0.0218.
. Solve the following equation: 4 (2y + 1) = 2(y - 13)(1 point)
GPT 3.5
To solve the equation 4(2y + 1) = 2(y - 13), we will first distribute the multiplication:
8y + 4 = 2y - 26
Next, we will isolate the y variable term by subtracting 2y from both sides:
8y - 2y + 4 = 2y - 2y - 26
This simplifies to:
6y + 4 = -26
Then, we can isolate the constant term by subtracting 4 from both sides:
6y + 4 - 4 = -26 - 4
This simplifies to:
6y = -30
Finally, we isolate the y variable by dividing both sides by 6:
(6y)/6 = -30/6
This simplifies to:
y = -5
Therefore, the solution to the equation 4(2y + 1) = 2(y - 13) is y = -5.
8y + 4 = 2y - 26
Next, we will isolate the y variable term by subtracting 2y from both sides:
8y - 2y + 4 = 2y - 2y - 26
This simplifies to:
6y + 4 = -26
Then, we can isolate the constant term by subtracting 4 from both sides:
6y + 4 - 4 = -26 - 4
This simplifies to:
6y = -30
Finally, we isolate the y variable by dividing both sides by 6:
(6y)/6 = -30/6
This simplifies to:
y = -5
Therefore, the solution to the equation 4(2y + 1) = 2(y - 13) is y = -5.
Q 10. Which equation is an identity?(1 point) Responses
GPT 3.5
To determine which equation is an identity, we need to find an equation that holds true for all values of the variable.
One example of an identity equation is:
x + 2 = x + 2
This equation is an identity because no matter what value we substitute for x, the equation will always hold true.
Therefore, the equation x + 2 = x + 2 is an identity equation.
One example of an identity equation is:
x + 2 = x + 2
This equation is an identity because no matter what value we substitute for x, the equation will always hold true.
Therefore, the equation x + 2 = x + 2 is an identity equation.
Q. 11.True or False? This equation has no solution: 12 x + 13 (x − 6) = 56 x + 2 (1 point)
GPT 3.5
To determine if the equation 12x + 13(x - 6) = 56x + 2 has no solution, we need to simplify and compare the variables on both sides of the equation.
First, let's distribute and simplify the equation:
12x + 13x - 78 = 56x + 2
Combining like terms, we get:
25x - 78 = 56x + 2
Next, let's isolate the x terms on one side of the equation and the constant terms on the other side:
25x - 56x = 2 + 78
Simplifying further:
-31x = 80
Finally, let's divide both sides of the equation by -31:
x = -80/31
Therefore, the equation 12x + 13(x - 6) = 56x + 2 does have a solution, namely x = -80/31.
The statement "This equation has no solution" is false.
First, let's distribute and simplify the equation:
12x + 13x - 78 = 56x + 2
Combining like terms, we get:
25x - 78 = 56x + 2
Next, let's isolate the x terms on one side of the equation and the constant terms on the other side:
25x - 56x = 2 + 78
Simplifying further:
-31x = 80
Finally, let's divide both sides of the equation by -31:
x = -80/31
Therefore, the equation 12x + 13(x - 6) = 56x + 2 does have a solution, namely x = -80/31.
The statement "This equation has no solution" is false.
Q 12. Two college friends rent an apartment. They have to pay the landlord two months rent and a $500 security deposit when they sign the lease. The total amount they pay the landlord is $2800. What is the rent for one month? Remember that they paid for two months plus the security deposit. Click the drop down menu to choose the correct amount.
GPT 3.5
To find the rent for one month, let's set up an equation based on the given information:
Let's denote the rent for one month as x.
They paid for two months of rent plus a $500 security deposit, so the total amount they paid is 2x + $500.
According to the question, the total amount they paid is $2800.
Setting up the equation:
2x + $500 = $2800
Now, let's solve for x:
Subtract $500 from both sides of the equation:
2x = $2800 - $500
2x = $2300
Next, divide both sides of the equation by 2:
x = $2300 / 2
x = $1150
Therefore, the rent for one month is $1150.
Let's denote the rent for one month as x.
They paid for two months of rent plus a $500 security deposit, so the total amount they paid is 2x + $500.
According to the question, the total amount they paid is $2800.
Setting up the equation:
2x + $500 = $2800
Now, let's solve for x:
Subtract $500 from both sides of the equation:
2x = $2800 - $500
2x = $2300
Next, divide both sides of the equation by 2:
x = $2300 / 2
x = $1150
Therefore, the rent for one month is $1150.
Q. 13. Noah and Kate are shopping for new guitar strings in a music store. They pay a total of $36.00 for their purchases. Noah buys 2 packs of strings. Kate buys 2 packs of strings and a music book. The book costs $16. How much is one pack of strings? Click the drop down menu to choose the correct answer.(1 point)
GPT 3.5
Let x represent the cost of one pack of strings.
Noah buys 2 packs of strings, so his total cost for the strings is 2x.
Kate buys 2 packs of strings and a music book, which costs $16. So her total cost for the strings and the book is 2x + $16.
The total cost for Noah and Kate's purchases is $36.
Setting up the equation:
2x + 2x + $16 = $36
Combining like terms:
4x + $16 = $36
Subtracting $16 from both sides of the equation:
4x = $20
Dividing both sides of the equation by 4:
x = $5
Therefore, one pack of strings costs $5.
Noah buys 2 packs of strings, so his total cost for the strings is 2x.
Kate buys 2 packs of strings and a music book, which costs $16. So her total cost for the strings and the book is 2x + $16.
The total cost for Noah and Kate's purchases is $36.
Setting up the equation:
2x + 2x + $16 = $36
Combining like terms:
4x + $16 = $36
Subtracting $16 from both sides of the equation:
4x = $20
Dividing both sides of the equation by 4:
x = $5
Therefore, one pack of strings costs $5.
Q. 14. A hairdresser is deciding to open her own studio. She will pay $1200 per month in rent and will charge $45 per haircut. If rent is her only expense, how many haircuts does she need to do to make a profit? Click on the dropdown menu to choose the correct option. Be careful about rounding correctly!(1 point)
GPT 3.5
To determine how many haircuts the hairdresser needs to do to make a profit, we need to set up an equation based on the given information.
Let's denote the number of haircuts as h.
The income from the haircuts is calculated by multiplying the number of haircuts by the price per haircut, which is $45h.
The monthly expense for rent is $1200.
To make a profit, the income from haircuts needs to be greater than the expense for rent.
Setting up the equation:
$45h > $1200
Now, let's solve for h:
Divide both sides of the inequality by $45:
h > $1200 / $45
h > 26.67
The hairdresser cannot perform a fraction of a haircut, so she would need to do at least 27 haircuts to make a profit.
Therefore, she needs to do at least 27 haircuts to make a profit.
Let's denote the number of haircuts as h.
The income from the haircuts is calculated by multiplying the number of haircuts by the price per haircut, which is $45h.
The monthly expense for rent is $1200.
To make a profit, the income from haircuts needs to be greater than the expense for rent.
Setting up the equation:
$45h > $1200
Now, let's solve for h:
Divide both sides of the inequality by $45:
h > $1200 / $45
h > 26.67
The hairdresser cannot perform a fraction of a haircut, so she would need to do at least 27 haircuts to make a profit.
Therefore, she needs to do at least 27 haircuts to make a profit.