Solving a word problem with three unknowns using a linear... The sum of three numbers is 104. The first number is 6 less than the second. The third number is 3 times the second. What are the numbers?
Answers
Let the unknown numbers be x, y, and z where
• First number = x
,• Second number = y
,• Third number = z
If the sum of the three numbers is 104, then;
[tex]x+y+z=104[/tex]If the first number is 6 less than the second, hence;
[tex]\begin{gathered} x=y-6 \\ \end{gathered}[/tex]Also, if the third number is 3 times the second, then;
[tex]z=3y[/tex]Substitute equations 2 and 3 into equation 1 to reduce the variables to the function of "y" only:
[tex]\begin{gathered} x+y+z=104 \\ (y-6)+y+3y=104 \\ y-6+4y=104 \\ 5y-6=104 \\ 5y=104+6 \\ 5y=110\rbrack \\ y=\frac{110}{5} \\ y=22 \end{gathered}[/tex]Get the first number "x"
[tex]\begin{gathered} x=y-6 \\ x=22-6 \\ x=16 \end{gathered}[/tex]Get the third number "z"
[tex]\begin{gathered} z=3y \\ z=3(22) \\ z=66 \end{gathered}[/tex]Therefore the three numbers are 16, 22, and 66.
Related Questions
(b) You need to raise a total of $600 for your trip. There are 10 days before you leave. You estimate that youwill be able to sell 15 cans of soda per day. How much should you charge for each can?Set up the problem:Ix 1Answer:$per can
Answers
The Solution:
To raise a total $600 in 10 days with an estimate of selling 15 cans of soda per day.
To find the amount you should charge for each can is:
[tex]\frac{\text{ \$600}}{\text{ 10 days}}\times\frac{\text{ 1 day}}{\text{ 15 cans per day}}=\frac{600}{150}=\text{ \$4}[/tex]Therefore, the correct answer is $4 for each can of soda.
Write a quadratic function in standard form whose graph passes through the given points1. (-1,5), (0,3), (3,9)
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The Standard Form of a Quadratic Function:
[tex]\text{ y = ax}^2\text{ + }bx\text{ + c}[/tex]Using the given points (-1,5), (0,3), and (3,9), let's substitute each point to the equation.
At (-1,5):
[tex]\text{ y = ax}^2\text{ + bx + c }\rightarrow5=a(-1)^2\text{ + b(-1) + c}[/tex][tex]\text{ 5 = a - b + c}[/tex]At (0,3):
[tex]\text{ y = ax}^2\text{ + bx + c }\rightarrow3=a(0)^2\text{ + b(0) + c}[/tex][tex]\text{ 3 = c}[/tex]At (3,9):
[tex]\text{ y = ax}^2\text{ + bx + c }\rightarrow9=a(3)^2\text{ + b(3) + c}[/tex][tex]\text{ 9 = 9a + 3b + c}[/tex]We now get these equations:
5 = a - b + c ; 3 =c; 9 = 9a + 3b + c
Let's determine the value of a, b and c. We get,
Substituting 3 = c to 5 = a - b + c,
[tex]\text{ 5 = a - b + c }\rightarrow\text{ 5 = a - b + 3 }\rightarrow\text{ a - b = 2}[/tex][tex]\text{ b = a - 2}[/tex]Let's substitute 3 = c and b = a - 2 to 9 = 9a + 3b + c,
[tex]\text{ 9 = 9a + 3b + c }\rightarrow\text{ 9 = 9a + 3(a-2) + 3}[/tex][tex]\text{ 9 = 9a + 3a - 6 + 3 }\rightarrow\text{ 12a = 9 + 6 - 3 }\rightarrow\text{ 12a = 12}[/tex][tex]\text{ a = }\frac{12}{12}\text{ = 1}[/tex]Since a = 1, let's solve for the value of b which is b = a - 2.
[tex]\text{ b = a - 2 }\rightarrow\text{ b = 1 - 2}[/tex][tex]\text{ b = -1}[/tex]Since we've identified that a = 1, b = -1 and c = 3, let's substitute the values to the standard form of a quadratic function to be able to make the equation.
[tex]\text{ y = ax}^2\text{ + bx + c }\rightarrow y=(1)x^2\text{ + (-1)x + (3)}[/tex][tex]\text{ y = x}^2\text{ - x + 3}[/tex]Therefore, the quadratic function in a standard form whose graph passes through the given points (-1,5), (0,3), (3,9) is y = x^2 - x + 3.
Jennifer made a cake and brownies. Her cake dish was 8 inches long, 5 inches wide and 2 inches deep. Her brownie tin was 9 inches long, 4 inches and 2 inches deep. Which dish has a larger volume? By how much?
Answers
Answer:
The cake is larger by 8 in^3
Step-by-step explanation:
The volume equals the product of the length of all 3 sides, or length * height * width
The volume of the cake is 8in*5in*2in which equals 80 in^3
The volume of the brownie tin is 9in*4in*2in which is 72 in^3
Clearly, the cake has more volume since 80 in^3 > 72 in^3
It is greater by the brownie tin by 80 in^3 - 72 in^3, or 8 cubic inches.
Solve quadratic equations -x^2 +3x < 3
Answers
The given quadratic inequation,
[tex]-x^2+3x<3[/tex]Subtract 3 on both sides,
[tex]\begin{gathered} -x^2+3x-3<3-3 \\ -x^2+3x-3<0 \end{gathered}[/tex]Multiply both sides by -1 that results in a change of symbol of the inequality,
[tex]\begin{gathered} (-1)\cdot(-x^2+3x-3)>(-1)\cdot0 \\ x^2-3x+3>0 \end{gathered}[/tex]Use the method of completing squares,
[tex]\begin{gathered} (x)^2-2(x)(\frac{3}{2})+(\frac{3}{2})^2-(\frac{3}{2})^2+3>0 \\ (x-\frac{3}{2})^2-\frac{9}{4}+3>0 \\ (x-\frac{3}{2})^2+\frac{3}{4}>0 \\ (x-\frac{3}{2})^2^{}>-\frac{3}{4} \end{gathered}[/tex]Consider that the square of any real number is always positive, and hence greater than a negative number.
So we can observe that the result obtained above is valid for all real number values of 'x'.
Thus, the solution of the given inequality is the set of all real numbers,
[tex]x=(-\infty,\infty)[/tex]help meeeeeeeeeeeeeeeeeeeeeee
thank you
Answers
Answer:
a) 1835 feet.
b) 1323 feet.
Step-by-step explanation:
Substitute each value of t into the equation to find the height at that particular time.
When t = 2, P(t) = -16(2)^2 + 1899 ⇒ P(t) = -64 + 1899 ⇒ P(t) = 1835.
When t = 6, P(t) = -16(6)^2 + 1899 ⇒ P(t) = -576 + 1899 ⇒ P(t) = 1323.
Situation: A 40 gram sample of a substance that's used for drug research has a k-value of 0.1472. N = Noekt No = initial mass (at time t = 0) N = fhass at time t k = a positive constant that depends on the substance itself and on the units used to measure time t = time, in days
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t = time, in days
k = 0.1472
N = 40 gram
[tex]40=N_0e^{(0.1472)t}[/tex]Connect the points to form a rectangle. What is the perimeter of the rectangle
Answers
The answer is option A 34 units.
We need to count the distance between the points and then add them to get the perimeter.
L and M are 5 units away, as well as X from J
L and X are 12 units away, as well as M from J
Now we can add the distances:
5 + 5 + 12 + 12 = 34
THen the perimeter is 34 units
Choose the coefficient closest to 0.Choose the coefficient with the least value.
Answers
Explanation:
In the given graph f(x) = a |x| .
The coefficient ''a'' increases the graph of the function will shrink and it will goes to zero.
And when coefficient ''a'' decreases the graph will expands.
The graph in the option C represents the graph below x-axis and the coefficient of |x| will be negative so it will be least value.
Answer: Option C.
The equation y=4.9x represents a proportional relationship. what is the constant proportionately?
Answers
A Web music store offers two versions of a popular song. The size of the standard version is 2.3 megabytes (MB). The size of the high-quality version is 4.1 MB. Yesterday, the high-quality version was downloaded three times as often as the standard version. The total size downloaded for the two versions was 3796 MB. How many downloads of the standard version were there?
Answers
The number of downloads of the standard version is 949.
What is an equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
We have,
Two versions of a song.
S = Standard version size = 2.3 MB
H = High-quality size = 4.1 MB
Hgh quality size = 3 x Standard quality size
H = 3 x S
H = 3S ____(1)
The total size downloaded for the two versions was 3796 MB.
This can be written as:
H + S = 3796 ______(2)
From (1) and (2) we get,
3S + S = 3796
4S = 3796
S = 3796/4
S = 949
Now,
From (1),
H = 3S = 3 x 949 = 2847
The number of downloads of the standard version is 949
The number of downloads of the High-quality version is 2847
Thus,
The number of downloads of the standard version is 949.
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Write an equation of the circle with center (6,8) and radius 7
Answers
Let’s find the equation of a circle with
radius r = 7 and center (h,k) =(6,8).
By definition, an equation of the circle with center (h,k) and radius r is
[tex](x-h)^2\text{ }+\text{ (}y-k)^2\text{ = }r^2[/tex]This is called the standard form for the equation of the circle. Then, in our case, replacing the data we have, we should have:
[tex](x-6)^2\text{ }+\text{ (}y-8)^2\text{ = 7}^2\text{ = 49}[/tex]so, we can conclude that the equation of the circle with center (6,8) and radius 7 is :
[tex](x-6)^2\text{ }+\text{ (}y-8)^2\text{ }^{}\text{ = 49}[/tex]If cost of 15 eggs is ? 75, then find out the cost of 4 frozen eggs
Answers
Problem:
If the cost of 15 eggs is $75, then find out the cost of 4 eggs?
Solution:
If the cost of 15 eggs is $75, then each egg cost:
[tex]\frac{\text{\$75}}{15}=\text{ \$5 for each egg}[/tex]thus, 4 eggs cost :
$5 x 4 = $20
then, the correct answer is $20
.name three fractions that are equivalent to 2/5
Answers
Answer:
4/10, 20/50, and 40/100
Hope this is helpful
Raffi has forgotten to multiply numbers that contain decimals. To multiply 0.4 x 1.34, he ignores the decimals
and multiples 4 x 134 = 536. Explain how Raffi could use estimation to determine where the decimal
point should be placed.
Answers
Answer:
.536 would be the correct answer. You could estimate by rounding 1.34 to 1 and 0.4 is close to .5 or 1/2. What is 1/2 of 1 1/2 or .5. This would lead me to place the decimal before the 5.
Step-by-step explanation:
Find c if a=2.93 mi,b=3.95 mi and c=40.3 degrees C=Round your answer to 3 decimal places
Answers
Using the law of cosines, we have the following relation between the measures of this triangle:
[tex]c^2=a^2+b^2-2ab\cos C[/tex]Plugging the given values, we have:
[tex]c^2=(2.93)^2+(3.95)^2-2(2.93)(3.95)\cos40.3^o[/tex]Solving for c, we have our answer:
[tex]c\approx2.556[/tex]The measure of c is 2.556 miles.
What is a unit rate?
Answers
we have that
A unit rate is a ratio between two numbers, where the second quantity is one unit
Example
5/1 m/sec is the same that 5 m/sec
that means
speed is 5 meters per one second
2 example
$6 per oz
that means -----> the cost is $6 per one ounce
A new energy drink advertises 177 calories in for 12 ounces. How many calories are in 20 ounces of the drink? _ calories
Answers
We can use the rule of three to solve this question, first, we need to build our relation, remeber that we need to put calories on one side and ounces on the other:
[tex]\begin{gathered} 177\rightarrow12 \\ x\rightarrow20 \end{gathered}[/tex]Now we need to solve it for x, we can build it as a fraction
[tex]\frac{177}{x}=\frac{12}{20}[/tex]Now we can solve it for "x", let's do a cross multiplication:
Therefore
[tex]12x=177\cdot20[/tex]We can divide by 12
[tex]x=\frac{177\cdot20}{12}[/tex]Now we can just to the calculus and we will have the value of x, if we do it we get
[tex]x=295\text{ calories}[/tex]Therefore, there are 295 calories in 20 ounces of drink
list the transformations
Answers
Answer:
on what?
Step-by-step explanation:
I can find percentages and values using the 68-95-99.7 rule, z-scores, and the standard normal distribution.answer the questions only if you know the answer please
Answers
ANSWER and EXPLANATION
1) We want to find the z score for a student who had a GPA of 3.8.
To do this, we have to apply the formula for z score:
[tex]z=\frac{x-\mu}{\sigma}[/tex]where:
x = score/GPA
μ = mean
σ = standard deviation
Therefore, the z score for the student with a GPA of 3.8 is:
[tex]\begin{gathered} z=\frac{3.8-3}{0.6} \\ z=\frac{0.8}{0.6} \\ z=\text{1}.33 \end{gathered}[/tex]2) The z score is a measure of how far away a data point is from the mean, in other words, it is a measure of how many standard deviations a data point is above or below a mean.
We can find how many standard deviations there are in a z score of -2.4 by dividing -2.4 by the standard deviation (0.6):
[tex]\begin{gathered} \frac{-2.4}{0.6} \\ -4 \end{gathered}[/tex]Therefore, since the z score is negative, we can conclude that the student's GPA is 4 standard deviations below the mean value.
The probability the Shane will sink a foul shot is 80%. If Shane attempts 40 foul shots, what is the probability that he sinks at least 28 shots? Round your answer to the nearest whole percenta-96%b-90%c-95%d-80%
Answers
For this problem we have to calculate the probability that x is greater or larger than 28
[tex]\begin{gathered} P(x\ge28)=\Sigma P(x=i) \\ i\text{ goes from 28 to }40 \end{gathered}[/tex]For each i, the probability
[tex]P(x=i)=\frac{40!}{28!\cdot12!}\cdot0.8^i\cdot0.2^{40-i}[/tex]Now computing the sum:
[tex]P(x\ge28)=96[/tex]The answer is 96%
the cost of manufacturing and selling x units of a product is c-7x+11 and the corresponding revenue R is R - x^2 - 15 find the number of the units needed to earn below and above the btrak even value
Answers
We know:
Profit = Revenue - Cost
When Revenue is equal to Cost, we have the break-even point.
Let's equate revenue and cost:
[tex]\begin{gathered} R=C \\ x^2-15=7x+11 \end{gathered}[/tex]To find the value of x, we can take all terms to LHS (Left-Hand-Side) and use the quadratic formula. The process of finding x is shown below:
[tex]\begin{gathered} x^2-15=7x+11 \\ x^2-15-7x-11=0 \\ x^2-7x-26=0 \\ u\sin g\text{ quadratic formula,} \\ x=-2.7,9.7 \end{gathered}[/tex]We can't have a negative value, so we disregard x = -2.7
What we have is
x = 9.7
So,
For 10 units (and above) sold, we will have a profit.
For 9 units (and below) sold, we will incur a loss.
PLSSS HELPPPPP IM RLYYYYY CONFUSEDDDDDD
Answers
Answer:
Top Left: -6
Top Right: -x
Bottom Left: 12x
Bottom Right: 2x^2
Trinomial: 2x^2 + 11x - 6
Step-by-step explanation:
Hello! Let's help you with your question here!
So, box method. From your question, the process is pretty easy. It would require a bit more work when factoring a trinomial.
The equation in question is already factored which is very useful, now it's a matter of the values in the box. I am on the assumption that the two top boxes and the two left boxes are filled in for you and you just need to fill the middle (if that isn't the case, do let me know). To just fill the boxes in the middle, evaluate it like you would multiplying rows and columns. So, it would be as such:
6 x
-1 -6 -x
2x 12x 2x^2
In terms of simplifying, we use FOIL to expand it out into a trinomial and collect like terms. Therefore, it becomes:
[tex](x+6)(2x-1) = (x*2x)+(x*(-1))+(6*2x)+(6*(-1))[/tex]
[tex]= 2x^2 + (-x) + (12x) + (-6)[/tex]
[tex]=2x^2+11x-6[/tex]
Let me know if there are any questions and if it is wrong, do let me know!
I got stuck with my math homeworkI was trying to figure these equations for 2 HOURS STRAIGHT (i gave up )
Answers
SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Write the given equation
[tex]1-5\left(x-1\right)=2\left(2x-1\right)-x[/tex]STEP 2: Simplify the equation
Expand the two sides of the equation
[tex]\begin{gathered} 1-5(x-1)=2(2x-1)-x \\ 1-5x+5=4x-2-x \end{gathered}[/tex]STEP 3: Simplify both sides
[tex]\begin{gathered} -5x+6=3x-2 \\ \mathrm{Subtract\:}6\mathrm{\:from\:both\:sides} \\ -5x+6-6=3x-2-6 \\ -5x=3x-8 \\ \mathrm{Subtract\:}3x\mathrm{\:from\:both\:sides} \\ -5x-3x=3x-8-3x \\ -8x=-8 \\ \mathrm{Divide\:both\:sides\:by\:}-8 \\ \frac{-8x}{-8}=\frac{-8}{-8} \\ \\ \therefore x=1 \end{gathered}[/tex]Hence, the value of x is 1
try to write an equation to describe the relationship between the stage number N and the number of squares S.
Answers
Let's first identify the number of squares in each stage:
Stage 1: 7 squares.
Stage 2: 14 squares.
Stage 3: 21 squares.
If we notice the sequence, we notice that it increases by 7 per stage, therefore:
Stage N: N*S
Where S is always equal to 7.
Foe example, in the stage 4:
Stage 4: 4*7 = 28 squares.
The equation is:
7*N
1 - Determine whether the given function is even, odd, or neither.a. [tex]f(x) = x ^{3} \times x ^{2} - 4[/tex]b. [tex]f(x) = 4x^{2} + x^{4} [/tex]2 - Evaluate the piece wise function at the given value of the independent variable. (picture)
Answers
We have a piecewise function that we should evaluate based on the condition of the independent variable
It is the value of the independent variable that will dictate which of the two equations to use
We can see that the value of the independent variable given is -3
Mathematically, -3 is not greater than 3; so this means the value of the independent variable given falls within the domain of the second part of the piece-wise function
Hence, we are to evaluate f(x) = -(x + 1) at the point x = -3
Thus;
f(-3) = -(-3 + 1)
f(-3) = -(-2)
f(-3) = 2
Given that angle ABC and EBF form a right angle, if measurement of ABC = 30° and EBF
= 6x° what is the value of x?
Answers
We know that ABC and EBF are complementary, or add up to 90°. So:
ABC + EBF = 90°
ABC = 30°
EBF = 6x°
6x° + 30° = 90°
6x° + 30 - 30 = 90 - 30
6x° = 60°
6x°/6 = 60°/6
x° = 10°
there are 60 seventh graders in the school. This make up 15% of the student in the school. How many students are in the school,total?
Answers
60 seventh graders make up 15% of the students in the school.
The procedure to solve this problem will be to first, divide 60 by 15 to find how much people make up 1%, and then multiply that number of people by 100% to find the total of students.
Step 1. Divide 60 people by 15% to find how many people are 1% of the students:
[tex]\frac{60}{15}=4[/tex]4 students are 1% of the total students.
Step 2. Now, we multiply the number we just find in step 1 by 100% to find the total amount of students in the school:
[tex]4\times100=400[/tex]There are 400 students in the school.
Answer: 400
find the image of P(7,7) under a dilation with scale factor 1/2 and center of dilation (1,3)
Answers
Given,
The coodinates of the points are P(7,7).
The scale factor is 1/2.
The coordinates of center of dilation is (1,3).
In the operation described here, it is the vector (center of dilation→ similar point) that will get multiplied by a factor 1/2.
The vector from the centre (1,3) to point (7,7) has coordinates (7,7) - (1,3)
[tex]((7-1),(7-3))=(6,3)[/tex]Now, dilated the coordinates by the scale factor of 1/2 then,
[tex]\frac{1}{2}(6,3)=(3,\frac{3}{2})[/tex]Image of the point is at,
[tex]\begin{gathered} (3,\frac{3}{2})=(3+1,\frac{3}{2}+3) \\ =(4,\frac{9}{2}) \end{gathered}[/tex]Hence, the coordinates of the image is (4,9/2).
Find the standard form of the equation of the circle having the following properties:
Center at the origin
Containing the point (-3,8)
Answers
Answer: x^2 + y^2 = 73
Step-by-step explanation:
H, K would be the center
R would be the radius
H, K= 0,0
(x-0)^2 + (y-0)^2= r^2
x^2 + y^2= r ^2
To Find Radius, you find distance between Origin, and Point
[tex]D=\sqrt{(x2-x1)^{2}+(y2-y1)^{2} }[/tex]
X2, Y2= (-3, 8)
X1, Y1= (0,0)
[tex]D= \sqrt{((-3)-(0))^{2} + ((8-0))^2} \\D= \sqrt{(-3)^2 + (8)^2} \\D= \sqrt{9+ 64} \\D=\sqrt{73} \\D= 6\sqrt{2}[/tex]
R= 6[tex]\sqrt{2}[/tex]
x^2+y^2= (6[tex]\sqrt{2}[/tex])
or x^2+y^2= 73
Help mee pleasee!!
thank you <3
Answers
As a function, the cost C(x) of publishing the newsletter would be C(x) = 75.00 + 0.25x, where the number of copies printed is x.
What is a function?The function is defined as a mathematical expression that defines a relationship between one variable and another variable.
We have been given that the cost of producing a newsletter is an initial charge of $75.00 for formatting and editing plus $0.25 for printing each copy.
Let the number of copies printed x and the cost C(x) of printing the newsletter as a function
As per the given data, the function would be as:
C(x) = 75.00 + 0.25x
Therefore, the cost C(x) of printing the newsletter as a function expressed would be C(x) = 75.00 + 0.25x.
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