We will assume that we want to know if the system of equations is independent or dependent:
[tex]\begin{cases}5x-3y=10\text{ (1)} \\ 6y=kx-42\text{ (2)}\end{cases}[/tex]where k is a real number. We will try to find the solutions to the system, and we will try to give values to k for which the system becomes independent or dependent.
We will use substitution, we solve for the variable x on the first equation to obtain:
[tex]\begin{gathered} 5x-3y=10 \\ 5x=10+3y \\ x=\frac{10+3y}{5} \end{gathered}[/tex]And now we replace it onto the second equation:
[tex]\begin{gathered} 6y=k(\frac{10+3y}{5})-42 \\ 6y=\frac{10k+3ky}{5}-42 \\ 6y=\frac{10k+3ky-210}{5} \\ 30y=10k+3ky-210 \\ 30y-3ky=10k-210 \\ y(30-3k)=10k-210 \\ y=\frac{10k-210}{30-3k} \end{gathered}[/tex]And the value of x will be:
[tex]\begin{gathered} x=\frac{10+3(\frac{10k-210}{30-3k})}{5}=\frac{10+\frac{10k-210}{10-k}}{5} \\ =\frac{10(10-k)+10k-210}{5(10-k)} \\ =\frac{100-10k+10k-210}{50-5k} \\ =-\frac{110}{50-5k} \\ =-\frac{22}{10-k} \end{gathered}[/tex]This means that a solution of the system will be:
[tex]\begin{cases}x=-\frac{55}{10-k} \\ y=\frac{10k-210}{30-3k}\end{cases}[/tex]Now, for finding the values which make the system dependent. This happens when the lines have the same slope, this is, when:
[tex]\begin{gathered} \frac{-5}{-3}=\frac{k}{6} \\ \frac{5}{3}=\frac{k}{6} \\ 30=3k \\ 10=k \end{gathered}[/tex]We did the division of the opposite of the coefficient of x, over the coefficient of y. This means that the system will be independent for each value of k different than 10, and will be dependent for k=10.
Can you please help me out with a question
Find the midpoint of the segment with the given endpoints. (-9,7) and (-4,2)
We have the following:
The midpoint is:
[tex]m=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]replacing:
[tex]m=(\frac{-9-4}{2},\frac{7+2}{2})=(\frac{-13}{2},\frac{9}{2})[/tex]The midpoint of the segment is:
[tex](-6.5,4.5)[/tex]x = 8 is a solution for equation 3x = 27 true or false
ANSWER
False
EXPLANATION
The guven equation is:
3x = 27
For x to be a solution of the equation, the value of x must be such that the left and right hand sides of the equation must match.
So, for x = 8:
3(8) = 27
24 = 27
As we can see, the two sides do not match, so x = 8 is not a solution.
What can you tell about the means for these two months? (1 point)The mean for April is higher than October's mean.There is no way of telling what the means are.The low median for October pulls its mean below April's mean.O The high range for October pulls its mean above April's mean.
The box plot shows the distribution of the data classified in quartiles.
If we want to know about the mean of the data set, we can only that it will be located within the range of the data set.
It will be closer to the median as the distribution gets less skewed.
By looking at the box plot, we can not confirm that April's mean is higher than October's mean, as the plots are overlapped.
We can not also concluded about the relation between the spread of the data and the relation with the mean.
Then, the most appropiate conclusion from the options is "There is no way of telling what the means are".
Look at the graph below and use the vertical line test to determine whether or not the graph represents a function. The determine if it is a one-to-one function.positive cube root functionThis graph Answer represent a function.This graph Answer represent a one-to-one function.
By using the vertical line test with the given graph
Since a vertical line can intersect the graph in any position in only ONE point
Then the graph represents a function
To test if it is a one-one function, draw a horizontal line and check if it intersects the graph at any position in only ONE point or not
Since a horizontal line can intersect the given graph in only ONE point in any position, then
The graph represents a one-one function
Consider the following mapping:(a)948546Part A:Write a set of ordered pairs to represent the mapping.
For order pair
In the first block the value 9 maps to 4 So, (9,4)
The value 8 maps to 4 & 6 so, the pair is (8,4) & (8,6)
The value 4 maps to 5So, (4,6)
The order pairs are (9,4), (8,4) (8,6) & (4,5)
b) Domain are the set of input values
In this mapping the first block map to the second thus, the value at the first block is the initial valua and act as domain of the mapping
So, the Domain = {9,8,4}
c) Range is the set of all the possible outputs
In this mapping, the second block show the final values of the function, thus the second block act as a range of the mapping
So, the Range= {4,5,6}
d) The mapping doesnot represent the function as the mapping is not one-one
A function must staisfy one-one rulei.r for every value of domain thier exists only one value of range
Where in this mapping we have, the range 4 whose domain are 9 & 8 does they donot satisfy the one-one creteria hence they are not one-one and niether they are function
Use arguments based on the Pythagorean theorem, its converse, and similar triangles to show that a triangle with sides 5n, 12n, and 13n is a right triangle. HINT: Start with n= 1, which results in side lengths of 5, 12, and 13. Answer in complete sentences and include all relevant calculations and algebraic manipulations
Sides:
5n
12n
13n
If n is 1 the triangle have sides: 5, 12, 13
The converse of the pythagorean theorem is: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle:
[tex]\begin{gathered} 13^2=12^2+5^2 \\ 169=144+25 \\ 169=169 \end{gathered}[/tex]Then, a triangle with sides 5, 12, 13 is a right triangle.
If n is 2 the triangle have sides: 10,24, 26
converse of the pythagorean theorem:
[tex]\begin{gathered} 26^2=24^2+10^2 \\ 676=576+100 \\ 676=676 \end{gathered}[/tex]Triangle with n=1 and n=2 are similars as the ratio between corresponding side is equal:
[tex]\begin{gathered} \frac{5}{10}=\frac{1}{2} \\ \\ \frac{12}{24}=\frac{1}{2} \\ \\ \frac{13}{26}=\frac{1}{2} \end{gathered}[/tex]Then, with any value of n the triangles are similar triangles.
Similar triangles have different sizes but the correpondign angles are the same (are congruent).
As triangles with sides 5n, 12n, 13n are similar triangles (All of then have the same measure on his correspondig angles) and makes true the Pythagorean theorem, they are right triangles.
I want to know the volume of the largest cube she could build with them.
All of the sides of a cube are equal, then, the volume is given by the cube of the length of any side.
We need to find the biggest cubic value smaller than 80.
[tex]\begin{gathered} 3\times3\times3=27 \\ 4\times4\times4=64 \\ 5\times5\times5=125 \end{gathered}[/tex]The largest cube has volume 64 cubic units, and the sides are 4 units long.
Tom goes fishing with Jason. Tom catches five trout and four catfish. Jason catches twice as many trout as Tom did. How many trout did Jason catch?
We know that Tom catches five trout and four catfish and we also know that Jason catches twice as many trout as Tom did.
Knowing that Jason catches twice as many trout as Tom did we must multiply the number of trouts that Tom caught (5 trouts) by 2
[tex]5\cdot2=10[/tex]So, Jason caught 10 trouts.
4) The capacity of a bathtub is 297 liters. The capacity of a sink is 9 liters, How many sinks of water will fill the bathtub? A 2,673 B 30 33 5) There are 354 milliliters of soda in each can. How much soda is there in cans? A 59 L B 2 L 124 mL © 360 L Short Answer Write the answer in the space given.
We are told that one can of soda has a capacity of 354 milliliters, then in order to calculate how much soda is there in 6 cans, we just have to multiply 354 mlliliters by 6, then we get:
Soda in 6 cans = 354 × 6 = 2124
We can split these 2124 milliliters as 2000 milliliters+ 124 milliliters.
1 liter is equivalent to 1000 milliliters, then we can convert the first 2000 milliliters to liters by dividing by 1000, then we get:
soda in 6 cans = 2000 ÷ 1000 liters + 124 milliliters
soda in 6 cans = 2 liters + 124 milliliters
Then, the amount of soda in 6 cans is 2liters 124milliliters, the correct answer is option B
Graph the function. f(x) = -3 sin x Use 3.14 for Use the sine tool to graph the function. The first point m value on the graph closest to the first point....I'll send pic of the problem
This is an example of how the graph should look like. What you can do is to find the sine of 2 angles, let's choose pi and 3pi/2. Find the function for these values
[tex]\begin{gathered} -3\cdot\sin \pi=-3\cdot0=0 \\ -3\cdot\sin (\frac{3\pi}{2})=-3\cdot-1=3 \end{gathered}[/tex]Plot these points and graph the function: (3.14,0) and (4.71,-3)
Jennifer got a new puppy and took him for a vet visit the vet said the puppy weighs 14 lb and only at 20% of it's adult weight. how much will the puppy weigh once its a an adult
PLS HELP 5 MATH QUESTIONS WILL MARK BRAINLIEST
The function f(x) = x³ - 2x is an odd function.
From the question, we have
f(x) = x³ - 2x,
The function to be odd if f(-x) = -f(x)
put x = -x in the function,
f(-x) = (-x)³ - 2(-x)
f(-x) = -x³ + 2x
Therefore, the function f(x) = x³ - 2x is an odd function.
Multiplication:
Mathematicians multiply the numbers to find the sum of two or more. It is a fundamental mathematical operation that is frequently employed in daily life. When we need to combine groups of similar sizes, we multiply. The fundamental concept of repeatedly adding the same number is represented by multiplication. The product of two or more numbers is the result of the multiplication of the factors, which are the amounts being multiplied. It is easier to repeatedly add the same number when the numbers have been multiplied.
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Carol Wynne bought a silver tray that originally cost $135 and was advertised at 35% off. What was the sale price of the tray?The sale price was $(Type an integer or a decimal.)
Let:
Op = Original price
r = Percentage discount = 35% = 0.35
Sp = Sale price
We can find the sale price as follows:
[tex]\begin{gathered} Sp=Op-r\cdot Op \\ so: \\ Sp=135-0.35\cdot135 \\ Sp=135-47.25 \\ Sp=87.75 \end{gathered}[/tex]Answer:
$87.75
find the supplement of the angle 19 degrees
Supplementary angles are the ones that when you add them, the result is 180°.
Let's call the angle we are looking for "x", since it is supplementary to the angle of 19°, they add up to 180°:
[tex]x+19=180[/tex]From this equation, we can solve to find the supplementary angle x.
We solve for x by subtracting 19 to both sides of the equation:
[tex]\begin{gathered} x+19-19=180-19 \\ x=161 \end{gathered}[/tex]Answer: 161°
A farm let's you pick 3 pints of raspberries for $12.00.What is the cost per pint?How many pints do you get per dollar?
Step 1
Given;
[tex]3\text{ pints of raspberries = \$12}[/tex]Required; To find the cost per pint and how many pints you get per dollar.
Step 2
Find the cost per pint using the ratio below
[tex]\begin{gathered} \frac{3\text{ pints of raspberries}}{1\text{ pint of raspberries}}=\frac{\text{\$}12}{\text{\$}x} \\ \end{gathered}[/tex]where;
[tex]\text{\$x=cost per pint}[/tex][tex]\begin{gathered} 3x=12 \\ \frac{3x}{3}=\frac{12}{3} \\ x=\text{\$}4 \end{gathered}[/tex]Step 2
Find how many pints you get per dollar.
[tex]\begin{gathered} \frac{1\text{ pint of raspberries}}{x\text{ pints of raspberries}}=\frac{\text{\$}4}{\text{\$}1} \\ 1=4x \\ \frac{4x}{4}=\frac{1}{4} \\ x=0.25\text{ pints of raspberries } \end{gathered}[/tex]Hence, you will get 0.25 pints of raspberries per dollar
Solving systems by substituting Y= -3x+52x+y=6
Answer:
x=-1, y=9.
Explanation:
Given the system of equations
[tex]\begin{gathered} y=-3x+5 \\ 2x+y=6 \end{gathered}[/tex]Substitute y=-3x+5 into 2x+y=6.
[tex]\begin{gathered} 2x+y=6 \\ 2x+(-3x+5)=6 \\ 2x-3x+5=6 \\ -x=6-5 \\ -x=1 \\ x=-1 \end{gathered}[/tex]Next, we solve for y.
[tex]\begin{gathered} y=-3x+5 \\ =-3(-1)+5 \\ =4+5 \\ y=9 \end{gathered}[/tex]The solution to the system of equations are:
x=-1 and y=9.
Consider the quadrilateral FACT below.FTAсСDetermine which of the following pairs is an opposite angle.
For any shaè, two angles are opposite if they are across each other, these angles share no sides. To determine which angles are opposite to each other you have to draw the diagonals of the quadrilateral since they go from one opposite angle to the other:
The opposite angles are:
F and C
A and T
The first option is the correct one.
Given the following rule, describe the transformation. (x , y) ---> (x + 9, y - 2)
The given rule is
[tex](x,y)\to(x+9,y-2)[/tex]The transformations shown indicate translations if the original point.
Any value (k) added/subtracted to the x-coordinate of a point results in a horizontal movement.
If the value is added to the x-coordinate → the resulting movement is k units to the rigth.
If the value is subtracted to the x-coordinate → the resulting movement is k units to the left.
Any value (m) added/subtracted to the y-coordinate of a point results in a vertical movement.
If the value is added to the y-cordinate → the resulting movement is m units up.
If the value is subtracted to the y-coordinate → the resulting movement is m units down.
In the given rule, 9 units are added to the x-coordinate, which indicates a translation 9 units to the right.
And there are 2 units subtracted to the y-coordinate, which indicates a translation 2 units down.
A rectangular room is 5 meters longer than it is wide, and its perimeter is 30 meters. Find the dimension of the room
The dimensions of the rectangular room is 10 meters and 5 meters respectively.
What is the dimensions of the room?The perimeter of a rectangle = 2(length + width)
Let
Width of the room = w metersLength of the room = (w + 5) metersPerimeter of the room = 30 metersThe perimeter of the rectangular room = 2(length + width)
30 = 2{(w + 5) + w}
30 = 2(w + 5 + w)
open parenthesis
30 = 2w + 10 + 2w
collect like terms
30 - 10 = 4w
20 = 4w
divide both sides by w
w = 20/4
w = 5
Hence,
Width of the room = w meters
= 5 meters
Length of the room = (w + 5) meters
= (5 + 5)
= 10 meters
Therefore, the length and width of the room are 10 meters and 5 meters respectively.
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Solve the inequality for A:-3 (9 – 4A) > 3 (2A – 11).
Given the inequality
-3 (9 – 4A) > 3 (2A – 11).
expand
-27 + 12A > 6A -33
Collect like terms
12A - 6A > -33 +27
6A > -6
Divide both sides by 6
A > -1
A piece of cheese contains 34.9 g of fat per 100 g. Calculate the number of g of fat in a 30 g serving of this cheese. Give your answer in g correct
to one decimal place.
The number of grams of fat in 30 grams of the pack will be 11.9 grams.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that a piece of cheese contains 34.9 g of fat per 100 g. The number of grams of fat in a 30g pack will be calculated as,
100 g of pack ⇒ 34.9 g of fat
1 g of pack ⇒ 34.9 / 100 g of fat
30 g of pack ⇒ (34.9 x 30 ) / 100 g pf fat
30 g of pack ⇒ 11.9 g of fat
Therefore, the number of grams of fat in 30 grams of the pack will be 11.9 grams.
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Evaluate the expression y = -4y^2 + 6y + 9
The given expression is
[tex]\begin{gathered} y^2\text{ + 6y + 9} \\ We\text{ would substitute y = - 4 into the given expression. It becomes} \\ (-4)^2+6(-4)+9 \\ 16-24+9 \\ =\text{ 1} \end{gathered}[/tex]The answer is 1
The current, I, in an electrical conductor varies inversely as the resistance,R, of the conductor. The current is 5 amperes when the resistance is 882ohms. What is the current when the resistance is 428 ohms? Round youranswer to two decimal places if necessary.
ANSWER:
10.30 A
SOLUTION
I=k/R this is base on the definition of I is inversely proportional to R
we need to find the constant k
5=k/882
k=4410
substitute k and R value to get I
I=4410/428
I=10.30
I'm going to show u the picture of the question
First, we need to know the number that chose vanilla flavor
Total number of classmate = 300
percentage of those that love Vanilla = 100% - 14% - 42% - 18% = 26%
Number of people that love Vanilla = 26% of 300 =
[tex]=\frac{26}{100}\times300[/tex][tex]=78[/tex]Next, find the number of those that chose strawberry
Number of people that love Strawberry = 18% of 300
[tex]=\frac{18}{100}\times300[/tex][tex]=54[/tex]
The number that chose vanilla than strawberry is 78 - 54= 24
f(1) = 4
f(2)= 25
f(n) = f(n − 2). f(n − 1)
f(3)=
The value of f(3) is 100 when f(1)=4 and f(2)=25 for function f(n) = f(n − 2). f(n − 1)
What is a function?A relation is a function if it has only One y-value for each x-value.
Given,
f(1) = 4
f(2)= 25
f(n) = f(n − 2). f(n − 1)
We need to find the value of f(3)
plug in 3 as n
f(3)=f(3-2).f(3-1)
f(3)=f(1)f(2)
Now put values of f(1) and f(2)
f(3)=4.25
f(3)=100
Hence the value of f(3) is 100 when f(1)=4 and f(2)=25 for function f(n) = f(n − 2). f(n − 1)
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The points U(−1,9), V(−1,5), and W(8,9) form a triangle.Plot the points then click the "Graph Triangle" button. Then find the perimeter of the triangle. Round your answer to the nearest tenth if necessary.
Remember that the coordinates of the points are written in the form (x,y), the first entry represents the distance over the horizontal axis and the second entry represents the distance over the vertical axis.
Plot the given points on the coordinate plane:
The length of the segment UV is 4, and the length of the segment UW is 9. Since the triangle VUW is a right triangle, use the Pythagorean Theorem to find the length of the segment VW:
[tex]\begin{gathered} VW=\sqrt{UV^2+UW^2} \\ \\ =\sqrt{4^2+9^2} \\ \\ =\sqrt{97} \\ \\ \approx9.849 \end{gathered}[/tex]Add the lengths of all the segments to find the perimeter of the triangle:
[tex]\begin{gathered} P=UV+VW+UW \\ \\ =4+9.849...+9 \\ \\ =22.849... \\ \\ \approx22.8 \end{gathered}[/tex]Therefore, to the nearest tenth, the perimeter of the triangle is 22.8.
choose the fraction pair that is equivalent. 3/4 and 4/3, 4/5 and 8/20, 8/24 and 1/3, or 3/12 and 1/3
To find out if two fractions are equivalent or not, we multiply by a cross. That is, multiply the numerator of the first fraction with the denominator of the second fraction and multiply the denominator of the first fraction with the numerator of the second fraction and check that it gives us the same result. For example:
[tex]\begin{gathered} \frac{1}{3}\text{ and }\frac{2}{6} \\ 1\cdot6=3\cdot2 \\ 6=6 \end{gathered}[/tex]So, in this case, you have
[tex]\begin{gathered} \frac{3}{4}\text{ and }\frac{4}{3} \\ 3\cdot3\ne4\cdot4 \\ 9\ne16 \\ \text{ They are not equivalent fractions} \end{gathered}[/tex][tex]\begin{gathered} \frac{4}{5}\text{ and }\frac{8}{20} \\ 4\cdot20\ne5\cdot8 \\ 80\ne40 \\ \text{ They are not equivalent fractions} \end{gathered}[/tex][tex]\begin{gathered} \frac{8}{24}\text{ and }\frac{1}{3} \\ 8\cdot3=24\cdot1 \\ 24=24 \\ \text{They are equivalent fractions} \end{gathered}[/tex][tex]\begin{gathered} \frac{3}{12}\text{ and }\frac{1}{3} \\ 3\cdot3\ne12\cdot1 \\ 9\ne12 \\ \text{ They are not equivalent fractions} \end{gathered}[/tex]Therefore, the fraction pair that is equivalent is
[tex]\frac{8}{24}\text{ and }\frac{1}{3}[/tex]A box is filled with 12 red cards 2 blue cards and 4 green cards a card is chosen at random from the box what is the probability that the card is not blue write your answer as a fraction in simplest form
There are 12+2+4=18 cards in total. The probability to get a not-blue card is
[tex]\begin{gathered} P(\text{not blue) = 1 - P(blue)} \\ P(\text{not blue) =}1-\frac{2}{18} \end{gathered}[/tex]which gives
[tex]\begin{gathered} P(\text{not blue})=1-\frac{1}{9} \\ P(\text{not blue})=\frac{9}{9}-\frac{1}{9} \\ P(\text{not blue})=\frac{9-1}{9} \\ P(\text{not blue})=\frac{8}{9} \end{gathered}[/tex]then, the probability that the card is not blue is 8/9.
A motorboat travels 200 miles in 5 hours going upstream. It travels 260 miles going downstream in the same amount of time. What is the rate of the boat in still water and what is the rate of the current?
B= Speed of the boat in the water =46 miles per hour
A= Speed of the Current =6 miles per hour
Speed upstream: B-A
Speed Downstream: B+A
Distance= Speed x time
Upstream :
(B-A) 5 = 200
Downstream:
(B+a) 5 = 260
Simplify both equations:
5B-5A =200
5B+5A =260
Add both equations:
10B = 460
Solve for B
B= 460/10
B= 46 miles/hour
Replace B on any distance equation:
5B-5A =200
5(46)-5A=200
Solve for A
230-5A=200
230-200=5A
30=5A
30/5 =A
A= 6 miles/hour