Answers
Options A, B, and D each have exactly one solution. Option C has no solution because it simplifies to a contradiction.
To determine which of the given equations have exactly one solution, we will analyze each equation to see if it can be simplified or rearranged to form an identity (true for all values of [tex]\( x \))[/tex] or a contradiction (false for all values of [tex]\( x \))[/tex], or if it remains a valid equation with a unique solution for [tex]\( x \).[/tex]
An equation has exactly one solution if it can be simplified to the form [tex]\( ax = b \)[/tex] where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, and [tex]\( a \)[/tex] is not zero.
Let's analyze each option step by step:
Option A: [tex]\( -5x + 12 = -12x - 12 \)[/tex]
1. Add [tex]\( 12x \)[/tex] to both sides: [tex]\( -5x + 12x + 12 = -12 + 12x \).[/tex]
2. This simplifies to [tex]\( 7x + 12 = -12 \).[/tex]
3. Subtract [tex]\( 12 \)[/tex] from both sides: [tex]\( 7x = -24 \).[/tex]
4. Divide by
This equation has exactly one solution, [tex]\( x = -\frac{24}{7} \).[/tex]
Option B: [tex]\( -5x + 12 = 5x + 12 \)[/tex]
1. Add [tex]\( 5x \)[/tex] to both sides: [tex]\( -5x + 5x + 12 = 5x + 5x + 12 \).[/tex]
2. This simplifies to [tex]\( 12 = 10x + 12 \).[/tex]
3. Subtract \( 12 \) from both sides: [tex]\( 0 = 10x \).[/tex]
4. Divide by [tex]\( 10 \)[/tex]: [tex]\( x = 0 \).[/tex]
This equation has exactly one solution, [tex]\( x = 0 \).[/tex]
Option C: [tex]\( -5x + 12 = -5x - 12 \)[/tex]
1. Subtract \( -5x \) from both sides: [tex]\( 12 = -12 \).[/tex]
This simplifies to a contradiction since [tex]\( 12 \)[/tex] does not equal [tex]\( -12 \).[/tex]
Therefore, this equation has no solution.
Option D: [tex]\( -5x + 12 = 5x - 5 \)[/tex]
1. Add \( 5x \) to both sides: [tex]\( -5x + 5x + 12 = 5x + 5x - 5 \)[/tex].
2. This simplifies to [tex]\( 12 = 10x - 5 \)[/tex].
3. Add [tex]\( 5 \)[/tex] to both sides: [tex]\( 17 = 10x \).[/tex]
4. Divide by [tex]\( 10 \)[/tex]: [tex]\( x = \frac{17}{10} \)[/tex].
This equation has exactly one solution, [tex]\( x = \frac{17}{10} \)[/tex].
Based on this analysis, Options A, B, and D each have exactly one solution. Option C has no solution because it simplifies to a contradiction.
Related Questions
In the half-life function Q(t)=28550⋅((34)h)(th) Q ( t ) = 28550 ⋅ ( ( 3 4 ) h ) ( t h ) , what is the half-life, h h , if (34)h=12 ( 3 4 ) h = 1 2 ?
Answers
The half-life h is calculated to be approximately 2.41.
To determine the half-life h from the given half-life function:
[tex]\[ Q(t) = 28550 \cdot \left( \frac{3}{4} \right)^{\frac{t}{h}} \][/tex]
we need to solve for h in the equation:
[tex]\[ \left( \frac{3}{4} \right)^h = \frac{1}{2} \][/tex]
To solve for h , we take the natural logarithm (ln) of both sides of the equation:
[tex]\[ \ln \left( \left( \frac{3}{4} \right)^h \right) = \ln \left( \frac{1}{2} \right) \][/tex]
Using the logarithmic property [tex]\(\ln(a^b) = b \ln(a)\)[/tex], we get:
[tex]\[ h \ln \left( \frac{3}{4} \right) = \ln \left( \frac{1}{2} \right) \][/tex]
Now, solve for h :
[tex]\[ h = \frac{\ln \left( \frac{1}{2} \right)}{\ln \left( \frac{3}{4} \right)} \][/tex]
Calculate the natural logarithms:
[tex]\[ \ln \left( \frac{1}{2} \right) = \ln(0.5) \approx -0.6931 \\\[ \ln \left( \frac{3}{4} \right) = \ln(0.75) \approx -0.2877 \][/tex]
Now, divide these values to find h :
[tex]\[ h = \frac{-0.6931}{-0.2877} \approx 2.41 \][/tex]
So, the half-life h is approximately:
[tex]\[ h \approx 2.41 \][/tex]
Therefore, the half-life h rounded to the nearest hundredth is:
h = 2.41
What equation represents the proportional relationship displayed in the table? x=2,4,6,8 y=10,20,30,40 Enter your answer in the box to complete the equation. y = ? x
Answers
The answer is y = 5x.
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Answer:
y=5x
Step-by-step explanation:
I did the test
What is a reasonable distance between two cities? A. 200 km B. 200 m C. 200 cm D. 200 mm
Answers
Answer:
A would be best
Step-by-step explanation:
200 km is roughly 124.2 miles
200 m is about 0.12 miles
200 cm and 200 mm is way too small to be distances between cities because 200 cm is like 78.7 inches
200 mm is like 7.8 inches
124.2 miles or 200 km would be a reasonable distances between two cities.
Answer:
A. 200km is your answer
Step-by-step explanation:
8 15 27 blank 20 find the blank number
Answers
the value of blank is 23.
Given series is,
8,15,27,_,20
We have to find the blank number.
Let the no. be x.
Since on observing the above series we find that the ratio of 15 to 8 is equal to the ratio of 27 to 15.
Hence, [tex]\frac{15}{8} =\frac{27}{15}=1.8[/tex]
This follows for the remaining terms also,
So, [tex]\frac{x}{27} =\frac{20}{x}[/tex]
[tex]x^{2} =20\times27\\x=\sqrt{540} \\x=23.23[/tex]
since, 23.23 is not a whole number so we take here 23.
Hence the value of blank is 23.
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Mickey buys 5 lemons and 3 limes for 3.70.Minnie buys 15 lemons and 4 limes for 7.85. How much does one lime cost?
Answers
Answer:
The cost of one lime is $0.65
Step-by-step explanation:
Let
x ---> the cost of one lemon
y ---> the cost of one lime
we know that
Mickey
[tex]5x+3y=3.70[/tex] ----> equation A
Minnie
[tex]15x+4y=7.85[/tex] ---> equation B
Solve the system by elimination
Multiply by -3 both sides equation A
[tex]-3(5x+3y)=-3(3.70)[/tex]
[tex]-15x-9y=-11.10[/tex] ----> equation C
Adds equation C and equation B
[tex]-15x-9y=-11.10\\15x+4y=7.85\\--------\\-9y+4y=-11.10+7.85\\-5y=-3.25\\y=0.65[/tex]
therefore
The cost of one lime is $0.65
Ron and Hermione are lifting feathers into the air with magic. Hermione can lift a feather 3,525
centimeters (cm) into the air. That is 5 times as high as Ron can lift a feather.
How high can Ron lift a feather with his magic?
Answers
Answer:705 cm
Step-by-step explanation:
3525 divided by 5
=705
Ron’s feather flew 705 cm
Ron’s feather flew 705 cm with his magic.
What is the unitary method?The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value.
Hermione can lift a feather 3,525 centimeters (cm) into the air. 5 times as high as Ron can lift a feather.
So
We have to 3525 divided by 5;
3525 / 5
=705
Hence, Ron’s feather flew 705 cm with his magic.
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Fifty students in the seventh grade are trying to raise at least $2000 for sports supplies. They already raised $750. How much should each student raise in order to meet the goal? Write your answer as an inequality.
Answers
Answer:
x ≥ 25
Step-by-step explanation:
In the seventh grade, fifty students are trying for sports supplies to raise at least $2000. They already have $750.
Let us assume that each student has to raise $x more to meet the goal.
Then, a total of 50 students will raise $50x more.
Therefore, the inequality to solve for x will be
750 + 50x ≥ 2000
⇒ 50x ≥ 1250
⇒ x ≥ 25 (Answer)
Final answer:
Each seventh grade student should raise at least $25 in order to collectively meet the goal of raising $2000 for their sports supplies after having already raised $750.
Explanation:
The goal amount for the seventh grade sports supplies fundraiser is $2000. They have already raised $750, leaving them with $1250 still needed to reach their goal. If there are 50 students working on the fundraiser, we can set up an inequality to determine how much money each student should raise minimally to meet the goal of $2000. Here's the inequality:
Let x be the amount each student needs to raise.
50x + $750 ≥ $2000
To solve for x, we subtract $750 from both sides of the inequality:
50x ≥ $2000 - $750
50x ≥ $1250
Now divide both sides by 50 to find the amount each student should raise:
x ≥ $1250 / 50
x ≥ $25
Therefore, each seventh grade student should raise at least $25 to meet the $2000 goal.
What is the square root of the day of the month that Christmas falls on?
Answers
Answer:
5
Step-by-step explanation:
The day of the month Christmas falls on is 25th because Christmas falls on 25th December. The square root of 25 is 5 because 5 squared is 25
How do you graph 4x+2y=10
Answers
Step-by-step explanation:
2x+y = 5 (divided by 2)
If x= 0, y=5
If x= 1, y=3
If x=-1, y=7
So, coordinate points are (0,5),(1,3),(1,7) and so on.
Continue above explanation.
After , draw a graph on x,y axis .
Answer: I used a graphing calculator.
Step-by-step explanation:
A credit card issuer charges an APR of 19.66%, and its billing cycle is 30 days
long. What is its periodic interest rate?
O
A. 21.72%
OB. 1.62%
O C. 21.53%
O D. 1.22%
SUBMIT
Answers
Answer:
The periodic interest rate is 1.62 %
Step-by-step explanation:
Given:
Annual percentage rate = 19.66%
Time period = 30 days
To Find:
Periodic interest rate =?
Solution:
The periodic interest rate r is calculated using the following formula:
r = (1 + \frac{i}{m})^{\frac{m}{n}} - 1
Where,
i = nominal annual rate
n = number of payments per year i.e., 12 for monthly payment, 1 for yearly payment and so on.
m = number of compounding periods per year
The period interest rate per payment is integral to the calculation of annuity instruments including loans and investments.
Now substituting the values we get
[tex]r = (1 + \frac{19.66}{12})^{\frac{12}{12}} - 1[/tex]
[tex]r = (1 + \frac{19.66}{12})^1 - 1[/tex]
r = (1 +1.638 ) - 1
r = (2.638 ) - 1
r = 1.638 %
The area of a square is 144 square centimeters. Find the length of the diagonal. Write your answer in simplest radical form.
Answers
Answer:
[tex]12\sqrt{2}\ cm[/tex]
Step-by-step explanation:
step 1
Find the length side of the square
we know that
The area of a square is equal to
[tex]A=b^{2}[/tex]
where
b is the length side
we have
[tex]A=144\ cm^2[/tex]
substitute in the formula of area
[tex]144=b^{2}[/tex]
solve for b
square root both sides
[tex]b=12\ cm[/tex]
step 2
Find the length of the diagonal
Applying the Pythagorean Theorem
[tex]d^2=b^2+b^2[/tex]
see the attached figure to better understand the problem
substitute the given values
[tex]d^2=12^2+12^2[/tex]
[tex]d^2=288[/tex]
square root both sides
[tex]d=\sqrt{288}\ cm[/tex]
simplify
[tex]d=12\sqrt{2}\ cm[/tex]
On a number line, what number is 2/3 of the way from 7 to 13?
Answers
Answer:
That would be about 11.667
The number that is 2/3 of the way from 7 to 13 on a number line is 11. This is found by calculating 2/3 of the distance between the two numbers and adding it to the starting number.
To find the number that is 2/3 of the way from 7 to 13 on a number line, you first calculate the distance between the two numbers, which is 13 - 7 = 6. Then you take 2/3 of that distance, which is 2/3 * 6 = 4. Lastly, you add this result to the starting number, in this case, 7, giving you 7 + 4 = 11. Therefore, the number that is 2/3 of the way from 7 to 13 is 11.
What is the equation of a line that is parallel to y=5/6x-10 and passes through (12,8)?
Answers
Answer:
y-8=5/6(x-12)
Step-by-step explanation:
y-y1=m(x-x1)
y-8=5/6(x-12)
The required equation of the line is [tex]\rm y = \dfrac{5}{6}x-2[/tex].
Given that,
The equation of the line is,
[tex]\rm y = \dfrac{5}{6}(x-10)[/tex]
It passes through the point (12, 8).
We have to determine
The equation of the line is parallel to the given line.
According to the question,
The equation of the line is,
[tex]\rm y = \dfrac{5}{6}(x-10)[/tex]
The slope of the line [tex]\rm m_1[/tex] is 5/6.
If the two lines are parallel to each other then the slope of these lines is the same.
[tex]\rm m_1 = m_2\\\\\dfrac{5}{6} = \dfrac{5}{6}[/tex]
Therefore,
The equation of line passes through the point (12, 8) is,
[tex]\rm( y -y_1) = m (x-x_1)\\\\(y-8) = \dfrac{5}{6} (x-12)\\\\6 (y-8) = 5(x-12)\\\\6y-48=5x-60\\\\6y = 5x-60+48\\\\6y = 5x-12\\\\y = \dfrac{5}{6}x- \dfrac{12}{6}\\\\y = \dfrac{5}{6}x-2[/tex]
Hence, The required equation of the line is [tex]\rm y = \dfrac{5}{6}x-2[/tex].
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there are 13 items in a bakers dozen. If Natalie orders 9 bakers dozen of muffins, how many muffins did Matslie order?
Answers
Natalie ordered a total of 117 muffins.
Explanation:A bakers dozen is equal to 13 items. If Natalie orders 9 bakers dozen of muffins, we can calculate the total number of muffins by multiplying 13 by 9.
Number of muffins = 13 * 9 = 117
Therefore, Natalie ordered 117 muffins.
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at a clothing store, jackets are on sale for 30% off.sale tax is 5%.which expression will calculate the total cost of a jacket with an original price of x
Answers
The expression to calculate the total cost of a jacket with an original price of x is 0.7x + 5 % of 0.7x
Solution:
Given that at a clothing store, jackets are on sale for 30% off
Sales is 5 %
To find: Expression to calculate the total cost of jacket
Let "x" be the original price of Jacket
Price of jacket after offer = original price of Jacket - 30 % of original price
Price of jacket after offer = x - 30 % of x
Price of jacket after offer = [tex]x - \frac{30}{100}x = 0.7x[/tex]
The total cost of jacket is sum of price of jacket after offer and sales tax
total cost of jacket = price of jacket after offer + sales tax
total cost of jacket = 0.7x + 5 % of 0.7x
Thus the expression to calculate the total cost of a jacket with an original price of x is 0.7x + 5 % of 0.7x
The graph of y= -4x + 7 is:
Answers
b and d can be eliminated because there are multiple solutions.
answer: A, because it shows all solutions to the equation
Answer: A line that shows the set of all solutions to the equation
Step-by-step explanation:
Given :
y = -4x + 7
The graph is a linear graph with negative slope.
the variable x could be any number , this means there could be as many solutions as possible.
this means that option B is out already.
Also , the equation is a graph and not a point , so option C and D are also out.
Option A will definitely fit in.
What is the value of x in the equation 5x 3-4x2
Answers
Answer:
x^2(5x-4)
Step-by-step explanation:
5x^3-4x^2=x^2(5x-4)
What are the exact solutions of x2 − x − 4 = 0, where x equals negative b plus or minus the square root of b squared minus 4 times a times c all over 2 times a?
A: x = the quantity of negative 1 plus or minus the square root of 15 all over 2
B: x = the quantity of 1 plus or minus the square root of 15 all over 2
C: x = the quantity of 1 plus or minus the square root of 17 all over 2
D: x = the quantity of negative 1 plus or minus the square root of 17 all over 2
Answers
Answer:C: x = the quantity of 1 plus or minus the square root of 17 all over 2
Step-by-step explanation:
Answer:
The answer is option C: x = the quantity of 1 plus or minus the square root of 17 all over 2
Step-by-step explanation:
The general form of the quadratic equation is ax² + bx + c = 0
The general solution of the quadratic equation is:
[tex]x=\frac{-b \pm \sqrt{b^2-4ac} }{2a}[/tex]
The given equation is x² − x − 4 = 0
So, a = 1 , b = -1 and b = -4
Substitute with a , b and c at the formula of the general solution.
The solution of the given equation = [tex]\frac{-(-1) \pm \sqrt{(-1)^2 - 4 * 1 * (-4)} }{2*1}= \frac{1\pm \sqrt{1 +16} }{2}=\frac{1\pm \sqrt{17} }{2}[/tex]
Comparing the solution with the given options
So, the answer is option C: x = the quantity of 1 plus or minus the square root of 17 all over 2
plz help in the attachment below
Answers
Answer:
M(t) = 741·(1/2)^(t/5730)
Step-by-step explanation:
One way to write an exponential function is this:
value at time t = (initial value) · (multiplier over period)^(t/period)
Here, the initial value is given as 741 grams, and the time period is 5730 years. The multiplier over that period is 1/2, since half of the quantity remains after that time. The problem statement tells us that "value at time t" is M(t), so we have ...
M(t) = 741·(1/2)^(t/5730) . . . . . t in years; M(t) in grams
[tex] {x}^{2} - 1 < 18[/tex]
Answers
[tex]
x^2-1 < 18 \\
x^2 < 19 \\
x < \sqrt{19}
[/tex]
Hope this helps.
Write an linear equation that passes through (5,2) and (6,4)
Answers
Answer:
2x-y=8
Step-by-step explanation:
(6,4)-(5,2)=(1,2)
L: 2x-y=k
(5,2) -> L, k=8
2x-y=8
Find the value of X and y?
Answers
Answer:
y = 12 ; x = 14
Step-by-step explanation:
2x + 7 =3x - 7 { corresponding angles}
7 + 7 = 3x - 2x
14 = x
x = 14
12y + 1 + 3x - 7 = 180 {co interior angles}
12y + 1 + 3 * 14 - 7 = 180
12y + 1 + 42 - 7 = 180
12y + 36 = 180
12y = 180 - 36
12y = 144
y = 12
PLEASE SOMEONE HELP ME ON THIS QUESTION!!!
Answers
Answer:
[tex]\frac{7}{3}[/tex]
Step-by-step explanation:
Im going to assume a faction greater than one means an improper fraction.
Converting [tex]2 \frac{2}{6}[/tex] gets [tex]\frac{14}{6}[/tex] which simplified is [tex]\frac{7}{3}[/tex]
I need help am confused
Answers
Answer:
125, 119, 113, 107, 101
Step-by-step explanation:
All you have to do is keep subtracting 6 from the numbers.
125 - 6 = 119
119 - 6 = 113
113 - 6 = 107
107 - 6 = 101
Answer:
119,210,337,506,723
Step-by-step explanation:
x³-6 where x=5,6,7,8,9
5³-6=15-6=119
6³-6= 216-6=210
7³-6=343-6= 337
8³-6=512-6=506
9³-6= 729-6=723
I need help trying to find out what is 73% of 83?
Answers
Answer:
60.59
Step-by-step explanation:
73%=0.73
0.73*83=60.59
60.59 is the number you're looking for.
Hope this helps, if not, comment below please!!!!
A sequence is defined by the recursive formula f(n+1)=f(n)-2. If f(1)=18, what is f(5)?
Answers
Answer:
[tex]f(5)=10[/tex]
Step-by-step explanation:
A sequence is defined by the recursive formula [tex]f(n+1)=f(n)-2[/tex]
If [tex]f(1)=18,[/tex] then
for [tex]n=1: f(2)=f(1+1)=f(1)-2=18-2=16[/tex]
for [tex]n=2: f(3)=f(2+1)=f(2)-2=16-2=14[/tex]
for [tex]n=3: f(4)=f(3+1)=f(3)-2=14-2=12[/tex]
for [tex]n=4: f(5)=f(4+1)=f(4)-2=12-2=10[/tex]
Answer:
10!
Step-by-step explanation:
If (10,3) and (6,31) are two
anchor points on a trend line,
then find the equation of the
line.
Answers
Answer:
y = - 7x + 73
Step-by-step explanation:
y = mx + b m: slope b: y intercept
m = (y-y') / (x-x') = (31-3) / (6-10) = 28 / - 4 = -7
b = y - mx = 3 - (-7) x 10 = 73
equation: y = -7x + 73
check: 31 = -7 x 6 +73 = -42 + 73 = 31 (6,31)
What is the equation of the line that passes through the point (-1, -3) and has a slope of -5?
answers y = -5x -16
y = -5x -8
y = -5x +8
y = -5x + 16
Answers
Answer:
y = -5x -8
Explanation:
Find b in equation y = mx + b
Insert numbers to find b
-3 = -5· -1 + b
Step 1: Simplify both sides of the equation.
−3=(−5)(−1)+b
−3=5+b
−3=b+5
Step 2: Flip the equation.
b+5=−3
Step 3: Subtract 5 from both sides.
b+5−5=−3−5
b=−8
Insert into equation
y = -5x - 8
Let’s start by writing out the model equation for slope-intercept
y=mx+b
M: Slope
X: Variable
B: Y-intercept
Now we know that m is slope. Let’s fill in the slope provided to us, which is -5
y=mx+b
y=-5x+b
Now let’s solve for b. We’ll do this by taking the point given to us and substituting it into the equation and solving.
y=-5x+b
*substitute the y and x of the point*
-3=-5(-1)+b
*multiply -5 and -1*
-3=5+b
*subtract 5 on both sides*
-8=b
Now you know b! Let’s put this into our equation
y=mx+b
y=-5x+b
y=-5x-8
This is your answer! Hope this helps comment below for more questions :)
Please help! This is the third time I asked
Find k, the constant of proportionality, for the data in this table. Then write an equation for the relationship.
They equation needs to be in the form y=kx
K=
Equation:
Answers
To find K you will pick a value from the x and then the corresponding Y value. Then we will plug it into the equation that is given y=Kx
Here:
X=25 Y=160
160=k25
(Divide both sides by 25 to isolate K)
K=6.5
Then plug K value into equation
Y=6.5X
Then if you want to check if that is correct you can pick a different point on the table and plug that into the equation to see if you get the same value.
Check:
X=50
Y=6.5x50
Y=320
Jade uses 8 cups of flour to make 24 muffins at that rate how much flour will it take to make 30 muffins
Answers
Answer:
10 cups
Step-by-step explanation:
First, we need to find the unit rate. 24/8 equals to 3. 3 is the amount of flour per cupcake. 24 + 3= 27 + 3= 30. You added twice to get to 30. 8+2=10