A solution to a system of linear equations in two variables is an ordered pair that makes BOTH equations true.
When is the system of equations is said to be consistent?
If there is at least one combination of values for the unknowns that satisfies each equation in the system—that is, makes each equation hold true as an identity—then a system of equations, whether linear or nonlinear, is said to be consistent.
When is the system of equations is said to be non-consistent?
If there isn't a set of values for the unknowns that solves every equation, then the system of equations is said to be inconsistent.
Ex: Consider a system of linear equations:
x+3y=6
x=-3y+6 …..(1)
2x+8y= -12 …..(2)
Substitute equation (1) in (2)
2(-3y+6)+8y=-12
-6y+12+8y=-12
2y=-24
y=-12 ….(3)
Substitute equation (3) in (1)
x+3(-12)=6
x-36=6
x=42
Here, the order pair (x,y)=(42,-12) is the solution of the given linear equations and satisfy equations (1) and (2)
> 42+3(-12) = 42-36 .....From (1)
= 6 = RHS
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What is the length of the segment shown on the graph? (Label your answer "units".)
Using the above graph, we can calculate that the segment has a length of 8 units.
What is the definition of a line segment?A line segment in geometry has two different points on it that define its boundaries. A line segment is sometimes referred to as a section of a line that links two places. The difference between a line and a line segment is that a line has no endpoints and can go on forever in any direction.Describe a line segment example.In real life, a line segment may be a stick, a pencil, or a ruler. The sun's beams are an illustration of a segmentThe graph shows the segment and dimensions given at the end of the segment are -4 and 4.
Thus we can say that, the first half of the segment is having the length 4 units and the other half also has the same length of 4 units.
By, adding both these lengths we get,
4 + 4 = 8 units
Therefore, the length of the segment is 8 units which is computed using the given graph.
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-6.75 Natural, whole, integer, rational, irracional, real,
-6.75 is a rational number.
Let us understand about natural, whole, integer, rational, and irrational numbers one by one.
Natural number:
Natural numbers are numbers that start from 1 and end at infinity. These are non-negative numbers.
Whole number:
Whole numbers are numbers that start from 0 and end at infinity.
We can also natural numbers including 0 is known as whole numbers.
These are non-negative numbers.
Integer:
An integer is a number with no decimal or fractional part and it includes negative and positive numbers, including zero.
Rational number:
A rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p, and a non-zero denominator q.
Also, the decimal representation of the rational number is terminating and recurring.
Irrational number:
An irrational number is a number that can not be expressed as the quotient or fraction p/q of two integers, a numerator p, and a non-zero denominator q.
Also, the decimal representation of the irrational number is non-terminating and non-recurring.
The given number is -6.75.
The decimal form is terminating.
So, it is a rational number.
Thus, -6.75 is a rational number.
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PLEASE HELP ME OUT What is x^2 - 4
Answer: (x-2)(x+2) is the solution
Step-by-step explanation:
Use the sum-product pattern
Common factor from the two pairs
Rewrite in factored form
Answer:
(x-2)(x+2)
Step-by-step explanation:
points (-2,1) and (3,y) have a slope of -3. find the y coordinate of the point
y coordinate of the point is -14
Explanation:Slope = change in y/change in x
[tex]\begin{gathered} m\text{ = }\frac{y_2-y_1}{x_2-x_1} \\ m\text{ = -3} \end{gathered}[/tex]Since m is negative, it means the slope is negative.
[tex]\begin{gathered} It\text{ also means the 2nd y value will be smaller than the 1st y value to get negative slope} \\ x_1=-2,y_1=1,x_2=3,y_2\text{ = y} \\ m\text{ = }\frac{y-1}{3-(-2)} \end{gathered}[/tex][tex]\begin{gathered} -3\text{ = }\frac{y-1}{3+2} \\ -3\text{ = }\frac{y-1}{5} \\ -3(5)\text{ = y - 1} \\ -15\text{ = y - 1} \\ \end{gathered}[/tex][tex]\begin{gathered} -15\text{ + 1 = y} \\ y\text{ = -14} \\ y\text{ coordinate of the point is -14} \end{gathered}[/tex]The domain for f(x) and g(x) is the set of all real numbers. let f(x)=3x+5 and g(x)=x^2. find f(x) +g(x)
1) In this question, we are dealing algebra of functions. So, let's add them up.
2)
[tex]\begin{gathered} f(x)=3x+5 \\ g(x)=x^2 \\ f(x)+g(x)=3x+5+x^2 \\ (f+g)(x)=x^2+3x+5 \end{gathered}[/tex]Note that we wrote it in the standard form.
2) Hence, the answer is:
[tex](f+g)(x)=x^{2}+3x+5[/tex]a homeowner estimates that it will take him 16 days to roof his house. a professional roofer estimates that he could roof the house in 3 days. how long will it take if the homeowner helps the roofer?
If the homeowner helps the professional roofer they will do the job in 2.53 days
How to determine how long it takes to do the roofing jobLet the number of days to complete the job be x
The rate at which the homeowner will roof is x / 16
The rate at which the professional roofer will roof is x / 3
Adding the rates together gives
x / 16 + x / 3
19x / 48
The rate it will take them both to finish the job is
= 48 / 19
= 2.53 days
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ctg 30° + 2 csc 60° - 3 cos 30°
The value of the expression ctg 30° + 2 csc 60° - 3 cos 30° is 5√3/6
The expression is calculated using trignometic values
ctg 30° + 2 csc 60° - 3 cos 30°
1/tan 30 + 2/sin 60 - 3 cos 30
1/(1/√3) + 2/(√3/2) - 3 (√3/2)
√3 + 4/√3 - 3√3/2
LCM is 2√3
(√3(2√3) + 2(4) - 3√3(√3) )/2√3
(6 + 8 - 9)/2√3
5/2√3
5√3/6
The value of the expression ctg 30° + 2 csc 60° - 3 cos 30° is 5√3/6
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A supply company manufactures copy machines. The unit cost C (the cost in dollars to make each copy machine) depends on the number of machines made. If x machines are made in the unit cost is given by the function C(x)=x^2-520x+79,797. What is the minimum unit cost? Do not round your answer.
Answer:
The minimum unit cost is $12,197.
Explanation:
The cost function is given below:
[tex]C\mleft(x\mright)=x^2-520x+79,797[/tex]To find the minimum unit cost, first, find the derivative of C(x).
[tex]C^{\prime}(x)=2x-520[/tex]Next, set the derivative equal to 0 and solve for x.
[tex]\begin{gathered} 2x-520=0 \\ 2x=520 \\ x=520\div2 \\ x=260 \end{gathered}[/tex]Finally, substitute x=260 into C(x) to find the minimum cost.
[tex]\begin{gathered} C\mleft(x\mright)=x^2-520x+79,797 \\ \implies C(260)=(260)^2-520(260)+79,797 \\ =67600-135,200+79,797 \\ =12,197 \end{gathered}[/tex]The minimum unit cost is $12,197.
Beginning with the equation 27x=54, write the new equation produced by dividing both sides by 9
Answer:
3x=6
Step-by-step explanation:
27/9=3, 54/9=6 so answer is 3x = 6
Kehlani went into a movie theater and bought 6 bags of popcorn and 8 drinks, costing a total of $76. Madelyn went into the same movie theater and bought 7 bags of popcorn and 4 drinks, costing a total of $62. Determine the price of each bag of popcorn and the price of each drink.
The price of each bag of popcorn and the price of each drink are $6 and $5 respectively.
What is equation?Equation: A declaration that two expressions with variables or integers are equal. In essence, equations are questions and attempts to systematically identify the solutions to these questions have been the driving forces behind the creation of mathematics.
Given:
The no of popcorn bags bought by Kehlani = 6,
The no of popcorn bags bought by Madelyn = 7,
The no of drinks bought by Kehlani = 8,
The no of drinks bought by Madelyn = 4,
The costing by Kehlani = $76,
The costing by Madelyn = $62,
Let the price of the popcorn bag is x and the price of the drink is y, then,
According to the question statement,
6x + 8y = 76 and 7x + 4y = 62,
Solve the equation by the elimination method,
x = 6 and y = 5
Therefore, the price of each bag of popcorn and the price of each drink are $6 and $5 respectively.
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Figure T is the result of a transformation on Figure S. Which transformation would accomplish this? 6 -4 -3 2 3 Figure S Figure T -3 -4 -5
Let:
[tex]\begin{gathered} A=(1,-1) \\ A^{\prime}=(3,-2) \\ A\to(x+a,y+b)\to A^{\prime} \\ so\colon \\ 1+a=3 \\ a=3-1=2 \\ ---- \\ -1+b=-2 \\ b=-2+1=-1 \\ so\colon \\ A\to(x+2,y-1)\to A^{\prime} \end{gathered}[/tex]Therefore, the transformation T is:
A translation 2 units right and 1 unit down
Playing basketball carries a met value of 8 and you weigh 130lbs or roughly 59kg, approximately how many calories per minute would you burn?.
The average person burns 575-775 calories per hour in a game of basketball. If they are shooting baskets, they will burn 325-450 calories per hour.
What is Calories?
Calories are a measure of energy. Two main definitions of "calorie" are frequently used due to historical factors. The amount of heat required to increase the temperature of one kilogramme of water by one degree Celsius was the original definition of the large calorie, food calorie, as well as kilogramme calorie (or one kelvin). The amount of heat required to produce the same increase in one gramme of water was known as the small calorie or gramme calorie. As a result, 1000 small calories are equal to 1 large calorie.
How many calories do basketball players burn?
Calories expended per minute equal (MET x body weight in kg x 3.5) / 200
"MET" is a unit of measurement for the energy expended during a given period of physical exercise. The chart above shows the MET for each activity.The energy required to perform a task with a MET of 1 is roughly equivalent to what it would take to sit stationary at room temperature without actively digesting food.When compared to a task with a MET of 1, one with a MET of 2 requires twice as much energy. Ten times as much energy is expended on a work with a MET of 10 as on one with a MET of 1.Hence, the average person burns 575-775 calories per hour in a game of basketball. If they are shooting baskets, they will burn 325-450 calories per hour.
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1. Michelle used a standard deck of 52 cards and selected a card at random. Afterrecording the suit of the card she picked, she then replaced the card.SultOutcomeSpades 9Hearts11Clubs7Diamonds3Part A: Determine the empirical probability of selecting a heart.Part B: Determine the theoretical probability of selecting a heart.Part C: Determine the empirical probability of selecting a club or diamond.Part D: Determine the theoretical probability of selecting a club or diamond.
The card deck is a standard deck of 52 cards, with 13 cards in each suit.
For the experiment carried out by Michelle, we have the following information:
n(Spades) = 9
n(Hearts) = 11
n(Clubs) = 7
n(Diamonds) = 3
The total number of times she performed the experiment is
[tex]n(\text{Total) = 9+11+7+3 = 30}[/tex]The empirical probability will make use of the experimental results, while the theoretical probability will make use of the total possibilities.
PART A: Empirical Probability of selecting a heart.
Probability is calculated by
[tex]P(\text{outcome) = }\frac{n(outcome)}{n(total)}[/tex]Therefore, the probability is calculated as
[tex]P(\text{heart) = }\frac{11}{30}[/tex]PART B: Theoretical probability of selecting a heart.
This is calculated by
[tex]\begin{gathered} P(\text{heart) = }\frac{13}{52} \\ P(\text{heart) = }\frac{1}{4} \end{gathered}[/tex]PART C: Empirical probability of selecting a club or diamond.
To calculate the probability for two outcomes, A or B, the probability can be calculated by
[tex]P(A\text{ or B) = P(A) + P(B)}[/tex]Therefore, we will find the probability of getting a club and then a diamond.
[tex]P(\text{heart) = }\frac{11}{30}[/tex][tex]P(\text{diamond) = }\frac{3}{30}=\frac{1}{10}[/tex]Therefore, the probability of selecting a club or a diamond is
[tex]\begin{gathered} P(\text{heart or diamond) = }\frac{11}{30}+\frac{1}{10} \\ P(\text{heart or diamond) = }\frac{7}{15} \end{gathered}[/tex]PART D: Theoretical probability of selecting a club or a diamond
We will find the probability of getting a club and then a diamond.
[tex]P(\text{club) = }\frac{13}{52}=\frac{1}{4}[/tex][tex]P(\text{diamond) = }\frac{13}{52}=\frac{1}{4}[/tex]Therefore, the probability of selecting a club or a diamond is
[tex]\begin{gathered} P(\text{heart or diamond) = }\frac{1}{4}+\frac{1}{4} \\ P(\text{heart or diamond) = }\frac{1}{2} \end{gathered}[/tex]Tanya is printing a report. There are 100 sheets of paper in the printer, and the number of sheets p left after
t minutes of printing is given by the function p(t) = -8t + 100.
Part A
How long would it take the printer to use all 100 sheets of paper? Explain how you found your answer.
It will take 12.5 minutes by the printer to use all 100 sheets of paper as the function is p(t) = -8t + 100 where p is the number of sheets left after t minutes of printing.
What is function?A mathematical expression, rule, or law that establishes the relationship between an independent variable and a dependent variable (the dependent variable). The property that every input is related to exactly one output defines a function as a relationship between a set of inputs and a set of allowable outputs. A mapping from A to B will only be a function if every element in set A has one end and only one image in set B. Let A & B be any two non-empty sets. Four broad categories can be used to classify different types of functions. dependent upon element Function is a one-to-one relationship, a many-to-one relationship, onto function, one-to-one and into function.
Here,
p(t)=-8t+100
Tanya is printing a report. There are 100 sheets of paper in the printer, and the number of sheets of paper p left after t minutes of printing is given by the function .When all the 100 sheets printed , then the number of sheets will left to print =0.
Substitute p(t)=0 in the given function, to find the value of t,
0=-8t+100
8t=100
t=12.5 minutes
The function is p(t) = -8t + 100, where p is the number of sheets left after t minutes of printing, and it will take the printer 12.5 minutes to use all 100 sheets of paper.
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The daily cost of production in a factory is calculated using f(x)= 400+ 13x where x is thenumber of products made. Which set of numbers best defines the domain of f(x)?A) IntegersB) Positive real numbersC) Positive rational numbersD) Whole numbers
Recall that the domain of a function is the set of numbers to which the function is defined. That is, if you replace the value of the independant variable (normally represented as x), then the function will give another number as a result.
In this case, we are given the function 400+13x. Since x is the independant variable, to determine the domain we must think on what possible values the variable x can take. Since in this case x represents the number of products made, it is impossible that it takes negative values, since in real life you can't produce negative amounts of products.
So far, we know that x should be greater or equal to zero. Once again, in this context, it doesn't make sense that, for example, x=7.8, since this would mean that 7.8 number of products were made. Since x represents the number of products made, it can only take values as x=0,x=1, x=2, x=3 and so on.
This set receives the name of whole numbers.
If EG = 10 and EF = 4, thenFG = [?]
In the exercise we first formulate and then replace values to find an unknown;
It is important to first look at what data the exercise gives us and then place them in the correct way, it must coincide with what the graph shows us.
A restaurant uses 8 ¼ pounds of flour to make 6 cakes. Beatriz wants to use the same recipe. How many pounds of flour does she need for one cake?
If A restaurant uses [tex]8\frac{1}{4}[/tex] pounds of flour to make 6 cakes. Beatriz wants to use the same recipe, then [tex]1\frac{3}{8}[/tex] pounds of flour is used for one cake
What is Division?A division is a process of splitting a specific amount into equal parts.
Given,
A restaurant uses [tex]8\frac{1}{4}[/tex] pounds of flour to make 6 cakes
We need to find how many pounds of flour does she need for one cake
To find this we just need to divide [tex]8\frac{1}{4}[/tex] pounds by 6.
[tex]8\frac{1}{4}[/tex] is 33/4
33/4/6
33/24
11/8
[tex]1\frac{3}{8}[/tex] pounds
Hence [tex]1\frac{3}{8}[/tex] pounds of flour is used for one cake
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Help the question is attached
The correct option will be A that is t=(A-P)/Pr by the simplification of equation, "an equation is a mathematical statement that shows that two mathematical expressions are equal ".
What is equation?In its most basic form, an equation is a mathematical statement that shows that two mathematical expressions are equal. 3x + 5 = 14, for example, is an equation in which 3x + 5 and 14 are two expressions separated by a 'equal' sign. In mathematics, an equation is a relationship of equality between two expressions written on both sides of the equal to sign. 3y = 16 is an example of an equation. Some examples of important equations are:
Linear equationsQuadratic equationsCubic equationQuadratic equationsDifferential equationsParametric equationsHere,
A=P+Prt
A-P=Prt
t=(A-P)/Pr
The answer is A, which is t=(A-P)/Pr using the equation's simplified form. An equation is a mathematical statement that demonstrates the equality of two mathematical expressions.
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Solve for x:
2.50+1.50x=3.25+1.25x
Answer:
x = 3
Step-by-step explanation:
Multiply both sides by 100
2.5 · 100 + 1.5x · 100 = 3.25 · 100 + 1.25x · 100
Refine:
250 + 150x - 250 = 325 + 125x - 250
Simplify:
150x = 125x + 75
Subtract 125x from both sides:
150x - 125x = 125x + 75 - 125x
simplify:
25x = 75
Now all you have to do is Divide both sides by 25:
[tex]\frac{25x}{25}[/tex] = [tex]\frac{75}{25}[/tex]
Now Finally the last part Now just Simplify so therefore the answer is
x = 3
A contractor is pouring a rectangular concrete slab with dimensions of 16 feet by 30 feet. To ensure that the sides of the slab form 90° angles, how many feet should each diagonal measure?.
If each sides of the slab form 90° angles, The number of feet that
each diagonal measure is 34 feet.
How to find the diagonal feet?
Using Pythagoreans theorem formula to find the diagonal feet
D² =L² + W²
Where:
D = Diagonal
L= Length = 16 feet
W = Width = 30
Let plug in the formula
D² = 16² + 30²
D² = 256 + 900
D=√1156
D= 34 feet
Therefore the diagonal measure 34feet
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Consider the probability model with sample space (A,B,C} and P(A)=0.1, P(B)=0.1, P(C)=0.8.Then
The sum of all the possible events is 1, which means that all the events are independent of the others, in that case
P(A or C) = P(A)+P(C) = 0.1 + 0.8 = 0.9
P(B and C) = P(B)*P(C) = 0.1 * 0.8 = 0.08
Please assist me, oh great smart people of the internet!
The value of x in (5 + 2x)/3 < (2x + 2)/4 is x < - 7.
What are inequalities and their types?Inequality is a relation that compares two numbers or other mathematical expressions in an unequal way.
The symbol a < b indicates that a is smaller than b.
When a > b is used, it indicates that a is bigger than b.
a is less than or equal to b when a notation like a ≤ b.
a is bigger or equal value of an is indicated by the notation a ≥ b.
Given, Inequality is (5 + 2x)/3 < (2x + 2)/4.
Now, (5 + 2x)/3 - (2x + 2)/4 < 0.
{4(5 + 2x) - 3(2x + 2)}/12 < 0.
{20 + 8x - 6x - 6}/12 < 0.
(2x + 14)/12 < 0.
2x + 14 < 0.
x + 7 < 0.
x < - 7.
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Answer the questions below.(a) The perimeter of a rectangular parking lot is 314 m.If the width of the parking lot is 71 m, what is its length?Length of the parking lot: 0 m(b) The area of a rectangular pool is 8075 m.If the length of the pool is 95 m, what is its width?Width of the pool: m
We have to use the perimeter formula for rectangles
[tex]P=2(w+l)[/tex]Where P = 314 m and w = 71 m.
[tex]\begin{gathered} 314=2(71+l) \\ \frac{314}{2}=71+l \\ 71+l=157 \\ l=157-71 \\ l=86 \end{gathered}[/tex]Hence, the length is 86 meters.An isosceles trapezoid has perimeter of 70. Each of the congruent nonparallel sides has length 13, and the height of the trapezoid is 12. How long is the longest side?
The length of the longest side is 27 units, in the isosceles trapezoid.
What is isosceles trapezoid?
A four-sided shape called a trapezoid has two parallel lines on each side (normally the top and bottom sides). A trapezoid that has equal-length non-parallel sides (the legs) is said to be isosceles. Considering everything mentioned before, we may start learning how to calculate an isosceles trapezoid's perimeter.
Given, the congruent nonparallel sides has length 13.
he height of the trapezoid is 12.
the extra length of the larger side will be 2*x, as it will be extra portion in both side.
So x² = 13²-12²
x² = 169-144 = 25
x = √25 = 5
So the lentgh of the larger side is 10 unit larger than smaller side.
The perimer of the isosceles trapezoid will be P = sum of all sides
P = 2*13 + L + L+10
⇒70 = 26+10 +2L
⇒2L = 70 - 36
⇒2L = 34
⇒L = 17
The length of the larger side will be 17+10 = 27 units.
Hence the length of the longest side is 27 units.
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12 eggs for 3$ how much is the cost of each egg
Answer:
0.25
Step-by-step explanation:
3 / 12 = 0.25
Answer: 0.60
Step-by-step explanation:
Select the correct image.
Which image shows triangle A dilated by a scale factor of 2 with a center at the origin?
The triangle ABC undergoes a dilation of scale factor 2 centered at the origin.
Dilation About the Origin
Given a point P(x,y), its dilation about the origin by a scale factor of k maps the point to a point P'(kx,ky).
If we are given the dilated point, then we can find the original point by dividing by the scale factor.
Triangle ABC was dilated by a scale factor of 2 about the origin mapping it to triangle A'B'C' with vertices A'(-4,4), B'(-4,6), and C(2,4).
The coordinates of triangle ABC are:
A=(-4/2,4/2) = (-2,2)
B=(-4/2,6/2) = (-2,3)
C=(2/2,4/2) = (1,2)
The coordinates of ABC are respectively (-2,2), (-2,3), and (1,2).
Hence the answer is the triangle ABC undergoes a dilation of scale factor 2 centered at the origin.
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Answer:i'm pretty sure it's the one to the right of the example
Step-by-step explanation:
(24x3 − 14x2 + 20x + 6) ÷ (4x2 − 3x + 5) = Q +
R
4x2 − 3x + 5
Q =
R =
Answer:
Step-by-step explanation:
Dividend = 24x³ - 14x² + 20x + 6
HELP QUICKLY PLSSSSS
Answer:
5, 6, 7, 8
Step-by-step explanation:
substitute the values of x from the table into f(x)
f(1) = 1 + 4 = 5
f(2) = 2 + 4 = 6
f(3) = 3 + 4 = 7
f(4) = 4 + 4 = 8
the values of f(x) are 5, 6, 7, 8
how can I rewrite 0.000000417 in appropiate scientific notation?
We have the following:
the number of 0 represents the number of the exponent
as is the are zeros to the right, the exponent is negative
[tex]0.000000417=4.17\cdot10^{-7}[/tex]Therefore, the answer is:
[tex]4.17\cdot10^{-7}[/tex]Which of these pieces of data about a package delivered by the post office is considered categorical data?A. Surface areaB. WeightC. Volume D. Dropoff location
Answer
D. Dropoff location
Step-by-step explanation
Quantitative variables are any variables where the data represent amounts (e.g. surface area, weight, or volume of a package). Categorical variables are any variables where the data represent groups (for example classifications like cities or brands).