How many solutions y 3x 5 has?

Answer:

Step-by-step explanation:

it is actually in the textbook pg 119

Constant, variable, operator and terms are four types of expression.

A constant term in mathematics is a term in an algebraic expression whose value is fixed or cannot change since it lacks any variables that may be changed. As the component is not yet included in the new term, a constant term that has a constant applied as a multiplicative coefficient likewise indicates a constant term. The term identify themselves as constants because the phrase has been modified. 5 is a constant term, in the quadratic polynomial 2x^2 + 5.

Yes, negative sign have nothing to do with the definition of constant. -4 is still a constant.

There are four types of expression in mathematics:

1. Constant: any numeric value e.g. 6

2. Variable: any alphabetic/unknow value e.g. x,y etc.

3. Operators: +,-, *, / (addition, subtraction, division, multiplication)

4. Term: it can be constant, variable, constant multiplied by variable. It is separated by an operator.

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(i) 0 is the remainder when x³ + 3x² + 3x + 1 is divided by (x +1)

(ii) 27/8 is the remainder when x³ + 3x² + 3x + 1 is divided by (x - 1/2)

(iii) 1 is the remainder when x³ + 3x² + 3x + 1 is divided by x

(iv) -π³ + 3π² - 3π + 1 is the remainder when x³ + 3x² + 3x + 1 is divided by (x + π)

(v) -27/8 is the remainder when x³ + 3x² + 3x + 1 is divided by (5 + 2x)

The value remaining after division is known as the Remainder. After division, we are left with a value if a number (dividend) cannot be divided entirely by another number (divisor). The remaining is the name for this amount.

Let a be any real number and let p(x) be any polynomial with degree greater than or equal to one.

Now, if a polynomial p(x) is divided by (x - a) then the remainder is p(a).

Assume, p(x) = x³ + 3x² + 3x + 1

Now,

(i) The root of x + 1 = 0 is -1

or, p(-1) = (-1)³ + 3(-1)² + 3(-1) + 1

or, p(-1) = -1 + 3 - 3 + 1

or, p(-1) = 0

Hence by the remainder theorem, 0 is the remainder when, x³ + 3x² + 3x + 1 is divided by x + 1. We can also say that x + 1 is a factor of x³ + 3x² + 3x + 1.

(ii) The root of x - (1/2) = 0 is 1/2.

or, p(1/2) = (1/2)³ + 3(1/2)² + 3(1/2) + 1

or, p(1/2) = 1/8 + 3/4 + 3/2 + 1

or, p(1/2) = (1 + 6 + 12 + 8)/8

or, p(1/2) = 27/8

Hence by the remainder theorem, 27 / 8 is the remainder when x³ + 3x² + 3x + 1 is divided by x.

(iii) The root of x = 0 is 0

or, p(0) = (0)³ + 3(0)² + 3(0) +1

or, p(0) = 0 + 0 + 0 + 1

or, p(0) = 1

Hence by the remainder theorem, 1 is the remainder when  x³ + 3x² + 3x + 1 is divided by x.

(iv) The root of x + π = 0 is - π

or, p(-π) = (-π)³ + 3(π)² - 3π + 1

or, p(-π) = -π³ + 3π² - 3π + 1

Hence by the remainder theorem, (-π)³ + 3(π)² - 3π + 1 is the remainder when x³ + 3x² + 3x + 1 is divided by x + π.

(v) Now, the root of 5 + 2x = 0 is -5/2

or, p(-5/2) = [(-5/2)³ + 3(-5/2)² + 3(-5/2) + 1]

or, p(-5/2) = [(-125/8) + (75/4) + (-15/2) + 1]

or, p(-5/2) = (-125 + 150 - 60 + 8) / 8

or, p(-5/2) = (-185 + 158) / 8

or, p(-5/2) = -27/8

Hence by remainder theorem, -27/8 is the remainder when x³ + 3x² + 3x + 1 is divided by 5 + 2x.

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The complete question is as follows:

Find the remainder when x³ + 3x² + 3x + 1 is divided by

i) x + 1    ii) x - (1/2)    iii) x    iv) x + π    v) 5 + 2x

Answer: 56 – 41 = 15

Step-by-step explanation:

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion. the missing lengths of HF is 4 and missing length PR =84.

What are Similar Triangles?

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion .

Triangle QRS is similar to triangle FGH.

Let us find the length of unknown side.

By comparing the sides.

SR/QR=HF/HG

8/14 = x/7

Apply cross multiplication

14x = 56

Divide both sides by 14

x = 56/14

x = 4

HF = 4

8)

By properties of triangle we have

QP/BP=PR/PC

56/24= x/36

Apply cross multiplication

24x = 2016

Divide both sides by 24

x = 2016/24

x = 84

PR = 84

Hence the missing lengths of HF is 4 and missing length PR =84.

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Answer:

Step-by-step explanation:

To find the area of the base of the box, we need to determine the length, breadth, and height of the box. We can set up a system of equations to represent the given information:

breadth = 2/3 * length

height = 1/3 * length

volume = length * breadth * height

Substituting the second equation into the third equation, we get:

volume = length * breadth * (1/3 * length)

      = length^2 * (2/3) * (1/3)

      = (2/9) * length^2

      = 48

      = 2^4 * 3

Solving for length, we find that length = 6. Plugging this value back into the first equation, we find that breadth = 4.

Therefore, the area of the base of the box is length * breadth = 6 * 4 = 24 square meters.

Answer:

3

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{7 cm}\underline{Scalar projection}\\\\Scalar projection $=\dfrac{u \cdot v}{|u|}$\\\\where:\\ \phantom{ww}$\bullet$ $u$ is the vector being projected onto.\\\end{minipage}}[/tex]

Given:

Calculate the magnitude of vector u:

[tex]\implies |u|=\sqrt{(0)^2+(-1)^2}=1[/tex]

The scalar projection is the magnitude of the vector projection.

Therefore, the magnitude of the projection of vector v onto vector u is:

[tex]\implies \dfrac{u \cdot v}{|u|}=\dfrac{\langle0, -1\rangle \cdot \langle-5, -3\rangle}{1}=\dfrac{0 +3}{1}=\dfrac{3}{1}=3[/tex]

Solution of the equation using substitution method is given by following steps:

1. Get the value of one variable in the other variable form using one equation.

2. Substitute the value of that variable in the second equation and get the value of both the variables.

Let us consider the system of equation to explain the solution of substitution method:

x + y = 0   __(1)

2x + 3y = 6  ___(2)

Get the value of y in the x form and substitute the value of y in equation (2) we get,

y = -x

2x + 3(-x) = 6

⇒2x - 3x = 6

⇒ -x = 6

⇒ x = -6

Now substitute the value of x in equation (1) we get,

y = -( -6)

⇒ y = 6

Therefore, the solution of the equation using substitution method is given by following way:

1. Get the value of one variable in the other variable form using one equation.

2. Substitute the value of that variable in the second equation and get the value of both the variables.

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Answer:

3/7

Step-by-step explanation:

[tex]g(-2)=\log_{2}(1)-4=-4 \\ \\ g(5)=\log_{2}(8)-4=-1 \\ \\ \frac{g(5)-g(-2)}{5-(-2)}=\frac{-1-(-4)}{5-(-2)}=\frac{3}{7}[/tex]

Answer: Part A: 994 feet higher thanks to absolute value.

Part B: -86 feet.

Step-by-step explanation: math

We will use the concept of Simplification of Equations . The value of p in the given equation is 1.

An Algerbraic equation in algebra is any equation that can be transformed in standard form like the following:

ax+bx+c=0 or ap+bp+cp+d=0 it can be of any form.

With a 0 and a, b, and c are known numbers, x is an unknown value, and a. The equation is linear, not quadratic, if a = 0 and b 0. The coefficients of the equation, denoted by the numbers a, b, and c, are the quadratic coefficient, linear coefficient, and constant or free term, respectively.

The roots or zeros of the expression on the left-hand side of the equation are the values of x that fulfill the equation. There can only be only solution to an Algebraic equation.

6p=2(5-2p) is the given equation

Divide by 2 on both LHS and RHS of the equation

6p/2 = 2(5-2p)/2

⇒3p=5-2p

⇒5p=5

∴p=1

We will use the concept of Simplification of Equations . The value of p in the given equation is 1.

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So, the length of triangle side a in centimeters is 6.

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane.

We can use the fact that the ratio of corresponding sides in similar figures is constant to solve this problem.

Since the two triangles are similar, we know that the ratio of their sides is constant. Letting x represent the length of side a in centimeters, we have

(a/3) = (8/4)

We can solve this equation by multiplying both sides by 3 to get

a = (8/4) * 3

= 6

Therefore, the length of side a in centimeters is 6.

The answer is therefore 6.

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Since the two triangles are similar, we know that the ratio of their sides is constant. Letting x represent the length of side a in centimeters, we have

(a/3) = (8/4)

We can solve this equation by multiplying both sides by 3 to get

a = (8/4) * 3

= 6

Therefore, the length of side a in centimeters is 6.

The answer is therefore 6.

The median x˜ is the data value separating the upper half of a data set from the lower half.

Arrange data values from lowest to highest value

The median is the data value in the middle of the set

If there are 2 data values in the middle the median is the mean of those 2 values.

Median Example

For the data set 1, 1, 2, 5, 6, 6, 9 the median is 5.

For the data set 1, 1, 2, 6, 6, 9 the median is 4. Take the mean of 2 and 6 or, (2+6)/2 = 4.

Median Formula

Ordering a data set x1 ≤ x2 ≤ x3 ≤ ... ≤ xn from lowest to highest value, the median x˜ is the data point separating the upper half of the data values from the lower half.

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The inverse of f is a function which takes any value x and divides it by 4, then adds 7 to the result.

The inverse of a function is the opposite of the original function, meaning that it takes the output of the original function and turns it back into the input. In this case, the original function is f(x)=1/4x+7. To find the inverse, we need to solve for x in the equation, which can be done by subtracting 7 from both sides and then multiplying both sides by 4. This gives us x=(x+7)/4, which is the inverse of the original function. To put it simply, the inverse of f takes any output of the original function, adds 7 to it, and then divides it by 4 to get the original input.

example;

If f(x)=1/4x+7 and x=3, then

f(3) = (1/4)3 + 7

7.75

The inverse of f is f^-1(x) = (x + 7) / 4.

If f^-1(7.75) = (7.75 + 7) / 4, then

f^-1(7.75) = 14.75 / 4

3.6875

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The graph of the function is illustrated below and the domain of the function is [-2, 2].

The term domain in math is referred as the set of all inputs of the function

Here we have to find and sketch the domain of the function f(x, y, z) = ln(64 − 4x² − 16y² − z²)

In order to sketch the function, we have to use the graphing calculator, and following the listed steps to sketch the graph.

The first step is to label the graph with appropriate axis like x, y, and z.

Now we have to input the function f(x, y, z) = ln(64 − 4x² − 16y² − z²) on the input field and then click on generate.

Now, we get the graph for the function.

After that we have to adjust the limit for the given graph.

Then we finally get the resulting graph as follows.

As per the definition of domain, and with the help of the given graph, we have identified the value of the domain is [-2, 2].

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Given a system of two equations in two variables, solve using the substitution method.

Word problems that require us to write systems of linear equations have two unknown quantities and two different ways to relate them.

This means we need to write two linear equations, and each contains the two unknowns as variables. The Understanding linear relationships lesson details how to translate verbal descriptions into equations.

After we write the equations, we can solve the system using our preferred method(s) for solving systems of linear equations covered in the Solving systems of linear equations lesson.

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The gradient of the line whose equation is x = -4 is: undefined.

The graph of an vertical line is a line that cuts the x-axis at (a, 0), of which has an undefined slope because it has a change in y but has no change in x. That is, the run of the line is zero. This means the gradient of a vertical line is undefined.

The equation of a vertical line is given as x = a, where a is the point that the line intersects the x-axis at.

Given the equation, x = -4, it means the line is a vertical line.

Therefore, since the gradient of vertical lines are undefined, the gradient of x = -4 is also determined to be undefined.

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The commission the employee earns as a percentage of the total cost of the gym membership is 19%.

A percentage is a value or ratio that may be stated as a fraction of 100. If we need to calculate a percentage of a number, we should divide it's entirety and then multiply it by 100. The percentage therefore refers to a component per hundred. Per 100 is what the word percent means. It is represented by %.

In this case, the gym keeps $42 and the employee who sold it gets $8.

The commission the employee earns as a percentage of the total cost of the gym membership will be:

= 8 / 42 × 100

= 19%

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The name of the condition in which factors f and g are correlated and may not be independent is confounded

What is  Fractional factorial?

In statistics, fractional factorial designs are experimental layouts made up of a carefully selected subset of full factorial design's experimental runs.

In a fractional factorial experiment, confounded factors are those that cannot be varied independently of each other because they are correlated. Confounded factors can affect the interpretation of the results of the experiment, as it may not be clear which factor is responsible for a particular effect. To control for confounded factors, it may be necessary to include them both in the experimental design or to use additional statistical techniques to account for their correlated nature.

Confounded factors are also known as aliased factors.

Hence, Name of the condition in which factors f and g are correlated and may not be independent is confounded

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The slope of the given equation y = 10 - 4x is equal to -4.

As given in the question,

Given equation is equal to y = 10 - 4x

Standard form in slope intercept form for the equation  is given by :

y = mx  + c

'm' is representing the slope of the given equation

'c' is representing the value of the  y-intercept of the given equation.

Now after simplifying the given equation in the standard form we get,

y = 10 - 4x

⇒ y = -4x + 10 __(1)

Compare equation (1) with the standard form of slope intercept form we get,

Slope of the equation is equal to m = -2.

Therefore, the slope of the given equation y = 10 - 4x is given by m = -4.

The given question is incomplete , the complete question is:

What is the slope of the equation y = 10 - 4x ?

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Answer:

Step-by-step explain

∛12↔3³≈4x↑

The missing number in an integer array of 1 to 100 can be found by summing the numbers 1-100 and subtracting the sum of the given array. The difference between the two sums will be the missing number.

To find the missing number, first sum the numbers 1-100 to get the total sum. Then sum the given array and subtract this sum from the total sum. The difference between the two sums will be the missing number. This method works because the total of the array should be equal to the sum of 1-100 if all the numbers are present. By subtracting the sum of the given array from the total sum, any missing numbers will be accounted for in the difference.

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Answer:

4

Step-by-step explanation:

In mathematics, the square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2, because 2 x 2 = 4. The square root of 9 is 3, because 3 x 3 = 9.

The square root of 16 is 4, because 4 x 4 = 16. Therefore, the value of the square root of 16 is 4.

Answer:

Step-by-step explanation:

The standard acceleration due to gravity (or standard acceleration of free fall), sometimes abbreviated as standard gravity, usually denoted by ɡ0 or ɡn, is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. It is defined by standard as 9.80665 m/s2 (about 32.17405 ft/s2).

The result of the each of the following set is

A ∪ B = {1, 3, 5, 6, 7, 9}

A ∩ B = {3, 9}

A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9}

A ∩ C = {∅}

A - B = {1, 5, 7}

B - A = {6}

B U C = {2, 3, 5, 6, 8, 9 }

B ∩ C = {6}

The given values are

A = {1, 3, 5, 7, 9}

B = {3, 6, 9}

C = {2, 4, 6, 8}

Then find the each given terms in set roaster notation

Union of the set, intersection of the set  and the difference of the set are the basic operations of set

A ∪ B = {1, 3, 5, 6, 7, 9}

A ∩ B = {3, 9}

A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9}

A ∩ C = {∅}

A - B = {1, 5, 7}

B - A = {6}

B U C = {2, 3, 5, 6, 8, 9 }

B ∩ C = {6}

Therefore, all the given terms has been found

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The roots of a quadratic equation can be found by analyzing the shape of the parabola formed by the equation's graph. The number of roots is determined by the number of x-intercepts.

To find the nature of the roots of a quadratic equation using a graph, you can plot the graph of the equation and analyze the shape of the graph. The graph of a quadratic equation is a parabola which opens upward or downward depending on the sign of the coefficient of the x² term. If the coefficient is a positive number, the parabola opens upward and if the coefficient is a negative number, the parabola opens downward. The roots of the equation are the x-intercepts of the graph, and there may be zero, one, or two roots. If the graph intersects the x-axis at two distinct points, the equation has two real, distinct roots. If the graph touches the x-axis at one point, the equation has one real root. If the graph does not intersect the x-axis, the equation has no real roots.

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For solving algebra Determine the issue, Describe your knowledge, Plan beforehand, Execute the plan ,and Check to see whether the response makes sense.

One of the many different mathematical disciplines is algebra. Algebra, a common thread that runs through nearly all of mathematics, is, roughly speaking, the study of mathematical symbols and the rules for using these symbols in formulae. With the use of mathematical expressions, algebra is the area of mathematics that helps to depict situations or problems. We employ integers that have a fixed or definite value in algebra, such as 2, 7, 0.068, etc. In addition to using numbers, algebra also makes use of variables like x, y, and z.

to do algebra for beginners - Determine the issue, specify your knowledge, Plan, execute, and confirm that the solution makes sense.

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The two triangles are similar if all three sides of one triangle are in proportion to the three sides of another triangle.

SSS, which stands for "side, side, side," denotes that we have two triangles with the same number of sides on each pair of corresponding sides. Two triangles are comparable if they have three pairs of sides that are in the same ratio. According to the SSS test for triangle similarity, two triangles are considered similar if their third sides are proportionate to one another.

In our daily lives, the idea of comparable triangles is really useful. By measuring the shadows' lengths and then utilizing triangles that are comparable to them, we can determine an object's height, such as the height of a building or a tower. Compare the shortest sides first, followed by the longest sides, when applying the SSS Similarity Theorem. Two triangles are similar if their corresponding side lengths are proportionate to one another.

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The measure of each side of the isosceles right triangle with hypotnuse 2.5√2 cm is  2.5 cm   .

The Isosceles Right Triangle is a type of right triangle which consists of two equal length legs .

In the right isosceles triangle the measure of the base and the perpendicular is same , So , base = perpendicular

let the side of the isosceles right triangle be = x .

given that the measure of the hypotnuse is = 2.5√2 cm

In a right triangle , By Pythagoras Theorem ,

base + perpendicular = hypotnuse

x² + x² = (2.5√2)²

2x² = 6.25 × 2

2x² = 6.25 × 2

2x² = 12.5

x² = 12.5/2

x² = 6.25 ,

x =  2.5 cm   ,....by taking square root on both the sides .

Therefore , the measure of each side is  2.5 cm  .

The given question is incomplete , the complete question is

What is the measure of each side of the given isosceles right triangle with a hypotenuse that measure 2.5√2 cm ?

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It is an iterated function also known as hailstone problem and can be solved on iterated steps.

Since then, the 3X + 1 issue has taken many different shapes. It is among the most infamous puzzles that have never been solved. For more than 40 years, prizes have been offered for its resolution, but nobody has totally and properly done so.

The 3x+1 Conjecture states that this function can be iterated through multiple times to eventually produce the value 1 starting from any positive integer n.

Take a positive integer X and divide it by 2 if it is even. If it is odd, multiply it by 3 and add 1 to the result to form the conjecture 3X + 1. The ends operate repeatedly according to the aforementioned criteria, and after a finite number of repetitions, the final end is always 1.

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