Given:
The system of equations is,
[tex]\begin{gathered} 3x-y-z=2.\text{ . .. . . .(1)} \\ x+y+2z=4\text{ . . . .. . (2)} \\ 2x-y+3z=9\text{ . . . . . . .(3)} \end{gathered}[/tex]The objective is to solve the equations using the elimination method.
Explanation:
Consider the equations (1) and (2).
[tex]\begin{gathered} 3x-y-z=2 \\ \frac{x+y+2z=4}{4x+z=6} \\ \ldots\ldots\ldots.(4)\text{ } \end{gathered}[/tex]Now, consider the equations (2) and (3).
[tex]\begin{gathered} x+y+2z=4 \\ \frac{2x-y+3z=9}{3x+5z=13} \\ \ldots\ldots\text{ . . . .. (5)} \end{gathered}[/tex]On multiplying the equation (4) with (-5),
[tex]\begin{gathered} -5\lbrack4x+z=6\rbrack \\ -20x-5z=-30\text{ . . . . . .(6)} \end{gathered}[/tex]To find x :
On solving the equations (5) and (6),
[tex]\begin{gathered} 3x+5z=13 \\ \frac{-20x-5z=-30}{-17x=-17} \\ x=\frac{-17}{-17} \\ x=1 \end{gathered}[/tex]To find z :
Substitute the value of x in equation (6),
[tex]\begin{gathered} -20(1)-5z=-30 \\ -5z=-30+20 \\ -5z=-10 \\ z=\frac{-10}{-5} \\ z=2 \end{gathered}[/tex]To find y :
Now, substitute the values of x and z in equation (2).
[tex]\begin{gathered} x+y+2z=4 \\ 1+y+2(2)=4 \\ y=4-1-4 \\ y=-1 \end{gathered}[/tex]Hence, the value of x is 1, y is -1 and z is 2.
I need help with this now. She said that we were supposed to solve for the area of the composite structures inside the structures.
We can decompose the figure in structures such that we can calculate the area of each structure.
Then, we have 3 rectangles, two triangles and a semicircle. The semicircle has diameter equals to
[tex]\begin{gathered} d=43ft-10ft-9ft=24ft \\ \end{gathered}[/tex]Then, the radius is equal to r=12ft.
We divide the figure in 6 different structures:
Structures I and V are triangles, so their area is
[tex]\frac{b\times h}{2}[/tex]Structures II, III and IV are rectangles, so their area is
[tex]a\times b\text{.}[/tex]Structure VI is a semicircle, so the area is
[tex]\frac{\pi r^2}{2}\text{.}[/tex]All the areas are in squared feet.
Structure I (b=10, h=48-37=11)
[tex]\frac{10\times11}{2}=55[/tex]Structure II (a=37, b-10)
[tex]37\times10=370[/tex]Structure III (a=38-12=26, b=43-10-9=24)
[tex]26\times24=624[/tex]Structure IV (a=32, b=9)
[tex]32\times9=288[/tex]Structure V (b=9, h=40-32=8)
[tex]\frac{9\times8}{2}=36[/tex]Structure VI (r=12)
[tex]\frac{(3.14)(12)^2}{2}=226.08[/tex]Then, we can obtain the total area adding all the area of the structures.
[tex]55+370+624+288+36+226.08=1599.08[/tex]So, the total area is 1599.08 squared feet.
What is the equation of the line shown in the graph, in standard form ? please help me as soon as possible.
First find two points and locate its coordinate
(-2, 0) and (-3, -2)
x₁=-2
y₁=0
x₂=-3
y₂=-2
substitute the values into the formula below
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex][tex]y-0=\frac{-2-0}{-3+2}(x+2)[/tex][tex]y=\frac{-2}{-1}(x+2)[/tex][tex]y=2(x+2)[/tex]y= 2x + 4
Re-arrange
2x - y = -4
What is the equation of a quadratic function y = f(x) with two irrational zeros, -√h and √h, where h is a rational number?
We want to write the equation of a quadratic function with two irrational zeros, -√h and √h where h is a rational number.
As the zeros are irrational, we know that what is inside the square roots must be negative. Thus, h is less than zero.
Now we will write the equation, by remembering the factor theorem. We can write the polynomial as:
[tex]f(x)=(x-x_1)(x-x_2)[/tex]Where x₁ and x₂ are the roots of the function, in this case, -√h and √h. This means that f can be written as:
[tex]\begin{gathered} f(x)=(x-(-\sqrt[]{h}))(x-\sqrt[]{h}) \\ =(x+\sqrt[]{h})(x-\sqrt[]{h}) \\ =x^2-(\sqrt[]{h})^2 \\ =x^2-h \end{gathered}[/tex]This means that the polynomial with the irrational zeros -√h and √h is f(x)=x²-h, where h is a negative rational.
²
6: Find the equation of a polynomial with given zeros:
The given zeros are - 5/4, - 2/3, 2
We would find the factors
For x = - 5/4,
4x = - 5
4x + 5 = 0
For x = - 2/3,
3x = - 2
3x + 2 = 0
For x = 2,
x - 2 = 0
We would multiply the factors. We have
(4x + 5)(3x + 2)(x - 2)
By applying the distributive proerty of multiplication, we vae
(4x + 5)(3x + 2) = 12x^2 + 8x + 15x + 10
(4x + 5)(3x + 2) = 12x^2 + 23x + 10
(x - 2)(12x^2 + 23x + 10) = 12x^3 + 23x^2 + 10x - 24x^2 - 46x - 20
(x - 2)(12x^2 + 23x + 10) = - 24x^2 = 12x^3 + 23x^2 - 24x^2 + 10x - 46x - 20
= 12x^3 - x^2 - 36x - 20
The first option is correct
Can you please help me with this trig ratios problem?
The expression is given
[tex]\sin \theta=\frac{\sqrt[]{3}}{2}[/tex]To determine the value of angle in degree.
[tex]\theta=\sin ^{-1}(\frac{\sqrt[]{3}}{2})[/tex][tex]\theta=60^{\circ}[/tex]Hence the value of angle in degree is 60 degree.
To calculate the cost of painting his silo, a farmer must find its height. The farmer uses a cardboard square to line up the topand bottom of the silo as shown in the diagram below. Approximate the height of the silo, rounded to the nearest foot.6 ftfeet10 ft
Given:
A figure with measurement is given.
Required:
Find the height of the silo.
Explanation:
The given figure is:
Use the geometric man theorem in the right triangle ABD.
[tex]\begin{gathered} \frac{AC}{BC}=\frac{CD}{AC} \\ AC^2=BC\times CD \end{gathered}[/tex][tex]\begin{gathered} CD=BD-BC \\ =BD-6 \end{gathered}[/tex]Substitute the given values.
[tex]\begin{gathered} (10)^2=6\times(BD-6) \\ 100=6(BD-6) \\ (BD-6)=\frac{100}{6} \\ (BD-6)=\frac{50}{3} \end{gathered}[/tex][tex]\begin{gathered} BD=\frac{50}{3}+6 \\ =\frac{50+18}{3} \\ =\frac{68}{3} \\ =22.7\text{ ft} \\ \approx23\text{ ft} \end{gathered}[/tex]Final Answer:
The height of the silo is approximately 23 ft.
BEUse substitution to solve.f2x2 = 5 + y4y = -20 + 8x2Solve the first equation for y and substitute it into the second equation. The resulting equati4y = 16x2 - 608x2 - 20 = -20 + 8x22x2 = 5 + 2x2 - 5
Given that:
[tex]\begin{gathered} 2x^2=5+y \\ 4y=-20+8x^2 \end{gathered}[/tex]From the first equation,
[tex]y=5-2x^2[/tex]Substitute the obtained value of y into the second equation.
[tex]\begin{gathered} 4(5-2x^2)=-20+8x^2 \\ 20-8x^2=-20+8x^2 \end{gathered}[/tex]So, it has infintely many solutions.
Solutions are of the form:
[tex](x,y)=(x,5-2x^2[/tex]where x is any real number.
find the present value for the following future amount $9780 at 2.5% compounded semiannually for 11 years (do not round until the final answer. then round to the nearest cent as needed)
The compound interest formula is given by:
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]where P is the principal (the initial value), r is the interest rate in decimal form, n is the number of times the interest is compounded in a given time t.
In this case we know that the future amount is 9780, this means that A=9780. Furthermore, we know that r=0.025, n=2 (since the interest is compounded semiannually) and t=11. Pluging this values in the formula a solving for P, we have:
[tex]\begin{gathered} 9780=P(1+\frac{0.025}{2})^{2\cdot11} \\ P=\frac{9780}{(1+\frac{0.025}{2})^{2\cdot11}} \\ P=7441.29 \end{gathered}[/tex]Therefore, the present value of our investment is $7441.29.
factor this polynomial completely[tex] {x}^{2} - x - 20[/tex]
Explanation
Given the expression
[tex]x^2-x-20[/tex]Therefore;
[tex]\begin{gathered} x^2-x-20 \\ =x^2-5x+4x-20 \\ =x(x-5)+4(x-5) \\ =(x+4)(x-5) \end{gathered}[/tex]Answer:
[tex](x+4)(x-5)[/tex]Bobby traveled 29 mles to get to work. It took him 32 minutes to get to his destination What was his speed durng hs trip to work?
distance, D = 29miles
time, T = 32mins
[tex]\text{speed = }\frac{dis\tan ce}{time}[/tex]Therefore,
[tex]S=\frac{29}{32}=0.91miles\text{ per minutes}[/tex]Follow the instructions below.5Write a: a without exponents.D+ロロ5a. a=0хx5Fill in the blank.5wa = a
1. Without exponents:
[tex]a\cdot a^5=a\cdot a\cdot a\cdot a\cdot a\cdot a[/tex]2.
[tex]\begin{gathered} a\cdot a^5=a^1\cdot a^5 \\ =a^{1+5}=a^6 \end{gathered}[/tex]Find the distance between (-6, 1) and (2, 2). Round to thenearest hundredth.
The given points are (-6, 1) and (2, 2).
To find the distance between these points, we need to use the following formula
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]Replacing the given points, we have
[tex]d=\sqrt{(2-(-6))^2+(2-1)^2}=\sqrt{(8)^2+(1)^2}=\sqrt{64+1}=\sqrt{65}[/tex]Which is approximately 8.06.
Therefore, the distance between these two points is 8.06 units, approximately.A restaurant added a new outdoor section that was 5ft wide 9ft long. What is the area of the new outdoor section?
Given:
a.) A restaurant added a new outdoor section that was 5 ft wide 9 ft long.
Based on the given, Length and Width, presuming it's rectangular in shape, let's now determine the area:
[tex]\text{ Area = Length x Width}[/tex][tex]=\text{ 9 x 5}[/tex][tex]\text{Area = 45 ft}^2[/tex]Therefore, the area of the new outdoor section is 45 ft.²
1 How many students are cleaning 96 desks if each student cleans 4 desks? a) Write an equation using a symbol for the unknown. b) Find the answer to the question and show your reasoning.
Given data:
The numbers of desk clean by a students are 4.
(a)
The expression for the given statement is,
[tex]1\text{ s= 4 d}[/tex]Here, s is the student and d represents desk.
(b)
Multiply 24 on both sides.
[tex]\begin{gathered} 24(\text{ 1 s)=24(4 d)} \\ 24\text{ s =96 d} \end{gathered}[/tex]Thus, for cleaning 96 desks 24 students needed.
complete the following deductive proof of the triangle angle sum theorem
The triangle angle sum theorem states that sumof interior angles in a triangle equals 180 degrees.
Here the interior angles are ∠3, ∠4, and ∠5.
To prove that m∠3 + m∠4 + m∠5 = 180 degrees, we have the following:
∠1 and ∠2 and ∠3 from line CP: Definition of a straight angle
m∠1 + m∠2 + m∠3 = 180 degrees: Substitution
Line CP is parallel to line AB: Given
∠1 ≅ ∠4, ∠2 ≅ ∠5: If two parallel lines are cut by a transversal, alternate interior angles are congruent.
m∠1 = m∠4, m∠2 = m∠5: Definition of congruence
m∠4 + m∠5 + m∠3: = 180 degrees: Commutative property
m∠3 + m∠4 + m∠5 = 180 degrees: Angle Addition Postulate
A potter use 3/5 of pound of clay to make a bowl. How many bowls could potter make 10 pound of clay?
Let x be the number of bowls the potter could make 10 pound of clay
3/5 pound= 1 bowl
10 pound = x
cross-multiply
[tex]\frac{3}{5}x=10[/tex]Multiply both-side of the equation by 5/3
[tex]x=10\times\frac{5}{3}[/tex][tex]=\frac{50}{3}[/tex][tex]=16\text{ }\frac{2}{3}[/tex]What is the solution to the system of equations in the graph?A. There are no solutionsB. There are infinitely many solutionsC. (3, 7)D. (7, 3)
Given the system of equations in the graph below:
[tex]\begin{bmatrix}x+y=3 \\ x+y=7\end{bmatrix}[/tex]Isolate x for : x+y=3
[tex]y=3-x[/tex]substitue y = 3 - x
[tex]\begin{gathered} \begin{bmatrix}3-y+y=7\end{bmatrix} \\ 3=7 \end{gathered}[/tex]Therefore there are no solutions:
Hence the correct answer is Option A
For each ordered pair, determine whether it is a solution to the system of equations:7x - 4y = 8-2x+3y=7
Answer:
The solution to the system of equations is
(x, y) = (4, 5)
Explanation:
Given the pair of equations:
[tex]\begin{gathered} 7x-4y=8\ldots.\ldots..\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\text{.}\mathrm{}(1) \\ -2x+3y=7\ldots\ldots\ldots\ldots\ldots.\ldots\ldots\ldots\ldots\ldots\ldots\ldots\text{.}\mathrm{}(2) \end{gathered}[/tex]To know the solution to the system, we solve the equations simultaneously.
From equation (1), making x the subject, we have:
[tex]x=\frac{8+4y}{7}\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\text{.}(3)[/tex]Substituting equation (3) in (2)
[tex]\begin{gathered} -2(\frac{8+4y}{7})+3y=7 \\ \\ \text{Multiply both sides by 7} \\ \\ -2(8+4y)+21y=49 \\ -16-8y+21y=49 \\ -16+13y=49 \\ \\ \text{Add 16 to both sides} \\ -16+13y+16=49+16 \\ 13y=65 \\ \\ \text{Divide both sides by 13} \\ y=\frac{65}{13}=5 \end{gathered}[/tex]The value of y is 5
Using y = 5 in equation (3)
[tex]\begin{gathered} x=\frac{8+4(5)}{7} \\ \\ =\frac{8+20}{7} \\ \\ =\frac{28}{7} \\ \\ =4 \end{gathered}[/tex]The value of x is 4
Which of the following expressions are equivalent to 4 - (-5) + 0?Choose 3 answers:А 4 - (-5)B 4 + 5С 4 - (-5 + 0)D (4 – 5) + 0E 4 – (5 – 0)
ANSWER
A, B and C
EXPLANATION
To find the expression that is equivalent as the given one, we will first simplify it:
4 - (-5) + 0
4 + 5 + 0
=> 9
Note:
- * - = +
- * + = + * - = -
Now, we will do the same for the others:
A. 4 - (-5)
=> 4 + 5
=> 9
It is equivalent
B. 4 + 5
=> 9
It is equivalent
C. 4 - (-5 + 0)
=> 4 + 5 - 0
=> 9
It is equivalent
The question askes for three answers, so the answers are:
A, B and C
While Shopping Margaret bought a total of 10 items ( pants and t shirts) each pair of pants cost $53, each shirt is $27. She spent $374. How many pants and How many shirts.
Given that she bought a total of 10 items ( pants and t-shirts) each pair of pants cost $53, each shirt is $27. She spent $374, we have two linear equations as follows:
Let x be the number of pants and let y be the number of t-shirts, then:
[tex]53x+27y=374[/tex][tex]x+y=10[/tex]In order to find how many and shirts she bought, we have to solve this linear system of two equations and two variables:
[tex]53x+27y=374\rightarrow53x=374-27y\rightarrow x=\frac{374-27y}{53}[/tex][tex]\mathrm{Substitut}e\mathrm{\:}x=\frac{374-27y}{53}\text{ }into\text{ x + y = 10}[/tex][tex]\frac{374-27y}{53}+y=10[/tex][tex]\frac{374+26y}{53}=10[/tex][tex]\mathrm{Isolate}\:y[/tex][tex]\frac{53\left(374+26y\right)}{53}=10\cdot \:53[/tex][tex]374+26y=530[/tex][tex]374+26y-374=530-374\rightarrow26y=156\rightarrow y=6[/tex]then:
[tex]x=10-y=10-6=4[/tex]So, she bought 4 pants and 6 t-shirts.
The equation represents the proportional relationship between the money earned (p) and the number of hot dogs sold (h)P=2hWhat is the money in dollars earned for each hot dog sold.Please help!!!
Answer:
$2
Explanation:
Given the below equation;
[tex]P=2h[/tex]where P = money earned in dollars
h = number of hot dogs sold
From the question, we are asked to determine the money in dollars earned for each hot dog sold, i.e., find P when h = 1;
[tex]\begin{gathered} P=2(1) \\ P=2dollars \end{gathered}[/tex]Find the assessed value of a home with a market value of $88,000 if theassessment rate is 42%.
Given that the Assessment rate = 42%
Market value = $ 88,000
To get the assessed value, we will find 42% of $ 88, 000
[tex]\begin{gathered} \frac{42}{100}\text{ x \$ 88, 000} \\ \\ \frac{3696000}{100} \\ \\ =\text{ \$ 36,960} \end{gathered}[/tex]Assessed value = $ 36,960
A tank is full and holds 600 gallons of water. It is being pumped out at a rate of 30 gallons per hour. Let y be the number of gallons in the tank. Is it discrete or continuous? And what are they domain and range?
The volume of water in the tank can be expressed by the equation below:
[tex]y(t)\text{ = 600-30}\cdot t[/tex]Where y is the volume of the tank and t is the time elapsed since the tank started pumping water out.
A function is continuous when it is difined for all the values inside its domain. The domain of this function is all the positive real numbers, this is because the input represents time and there isn't negative time. For each fraction of hour there will be an associated volume of water in the tank, therefore the function is continuous.
The domain of the function is:
[tex](0,\infty([/tex]The range of the function is the set of values that it can output for the given domain. In this case the maximum value of the function is the total capacity of the tank 600 gallons, while the minimum value happens when the tank is empty, so the range is:
[tex](0,600)[/tex]Damon deposits $2,240 into a savings account that earns 3½% interest compounded quarterly. What is his new balance after 6 months?
Given:
Principal, P = $2,240
Interest rate, r = 3½% = 3.5% = 0.035
Time, t = 6 months = 6/12 months a year = 0.5 years
Yearly deposits, n = 4 (quaterly)
Use the compound interest formula below:
[tex]A\text{ = P(}1\text{ + }\frac{r}{n})^{nt}[/tex]Therefore, we have:
[tex]A\text{ = 2240(1 + }\frac{0.035}{4})^{4\cdot0.5}[/tex]Solving further,
[tex]A\text{ = 2240 (1 + }0.1757)[/tex][tex]\begin{gathered} A\text{ = 2240( 1.01757)} \\ \text{ = }2279.3715 \end{gathered}[/tex]Therefore his new balance after 6 months is $2279.37
ANSWER:
$2,279.37
the cost of a dozen carnations at the florist is 4 1/3 dollars. how much would it cost for 3 dozen?
1 dozen = 4 1/3 dls. = 13/3
3 dozen = (13/3)*12 = 52 dollars.
2 1/2 : 3 1/2 = n:2what do the : between numbers mean?
The symbol : denotes that the two ratios are in proportion
[tex]\begin{gathered} 2\frac{1}{2}\colon3\frac{1}{2}=n\colon2 \\ we\text{ can rewrite it as } \\ \frac{2\frac{1}{2}}{3\frac{1}{2}}=\frac{n}{2\text{ }} \\ In\text{ this form we can easily se}e\text{ that in order to solve for n , we just have to multiply both sides of the equation by 2} \\ 2\cdot\mleft\lbrace\frac{2\frac{1}{2}}{3\frac{1}{2}}=\frac{n}{2\text{ }}\mright\rbrace\cdot2 \\ =\frac{2\cdot2\frac{1}{2}}{3\frac{1}{2}}=n \\ n=\frac{2\cdot2\frac{1}{2}}{3\frac{1}{2}}=\frac{5}{3\frac{1}{2}}=\frac{5}{\frac{7}{2}}=5\cdot\frac{2}{7}=\frac{10}{7} \\ \\ \end{gathered}[/tex][tex]2\frac{1}{2}\colon3\frac{1}{2}=n\colon2[/tex]we can rewrite the expression as :
[tex]\frac{2\frac{1}{2}}{3\frac{1}{2}}=\frac{n}{2\text{ }}[/tex]Now we can see easily that that in order to solve for n , we just have to multiply both sides of the equation by 2
[tex]2\cdot\lbrace\frac{2\frac{1}{2}}{3\frac{1}{2}}=\frac{n}{2\text{ }}\rbrace\cdot2[/tex][tex]=\frac{2\cdot2\frac{1}{2}}{3\frac{1}{2}}=n[/tex][tex]n=\frac{2\cdot2\frac{1}{2}}{3\frac{1}{2}}=\frac{5}{3\frac{1}{2}}=\frac{5}{\frac{7}{2}}=5\cdot\frac{2}{7}=\frac{10}{7}[/tex]Juanita was asked to create a scale drawing of her bedroom. Using the dimensions and scale shown, determine the actual area of her room.
Here, we want to get the actual area of the room
The shape is rectangular and the area of the shape is simply the product of the dimensions of its sides
Now, what we have to do here is to have the actual dimensions
That means we will multiply each of the dimensions in inch by 5 to get the dimensions in ft
Mathematically, we have this as;
[tex]\begin{gathered} 2.2\text{ in }\times\text{ 5 = 11 ft} \\ 4.3\text{ in }\times\text{ 5 = 21.5 ft} \end{gathered}[/tex]The actual area is thus the product of the actual dimensions above
We have the actual area as;
[tex]11\text{ ft }\times21.5ft=236.5ft^2[/tex]know that two of the side lengths of the triangle are 3 inches and 4 inches. We represent the third side length of the triangle with the variable x.
Answer:
x < 7
Explanation:
Given the following sides of a triangle
s1 = 3
s2 = 4
Reqquired
Third side x (hypotenuse)
Using the pythagoras theorem;
x^2 = s1^2 + s2^2
x^2 = 3^2 + 4^2
x^2 = 9+16
x^2 = 25
x = \sqrt{25}
x = 5
Since x is equal to 5, this means we can say that the values of x is less than 7 (x < 7) based on the option
i need help with a non graded 10 question prep test
Given data:
The given triangles is shown.
The expression for the ratio of adjacent side is,
[tex]\frac{3}{6}=\frac{1}{2}[/tex]Thus, the scale factor is 1/2, so the second option is correct.
Hi I need help to make sure I'm answering this question right
Firstly, we are going to plot the given on a graph sheet.
The inequalities sign would be reverse. This is to ensure we have a clearer to ascertain where the solutions
This is shown in the image below:
The points that fall on the unshaded part of the graph are the solutions to the system of inequalities and these points are (-1,8) and (1,5).
Hence, the correct options are E and F