Writing an equation of a line given its slope and y-intercept
Write an equation in slope-intercept form for the line with slope -3 and y-intercept 4.

Answer:

x = pi/12, 5pi/12, 13pi/12, 17pi/12.

Step-by-step explanation:

Note that cos (4x) =  1 - 2sin^2 (2x)

Substituting we have:

1 - 2sin^2 (2x) - 9 sin(2x) + 4 = 0

2 sin^2 (2x) + 9 sin(2x) - 5 = 0

(2 sin 2x - 1 )(sin 2x + 5 ) = 0

sin 2x = 1/2 and sin 2x = -5.

sin 2x = 1/2,   gives 2x = pi/6, 5pi/6, 13pi/6, 17pi/6.

There are no solutions to sin 2x = -5  because the range of sin x  is -1 to +1.

So x = pi/12, 5pi/12, 13pi/12, 17pi/12.

To solve the trigonometric equation cos(4x) - 9 sin(2x) + 4 = 0, we can use trigonometric identities and algebraic manipulation. The steps to solve the equation are writing it as a quadratic equation, substituting with sin(x), solving the quadratic equation, and substituting back to solve for x.

To solve the trigonometric equation cos(4x) - 9 sin(2x) + 4 = 0 for 0 to 2pi, we can use trigonometric identities and algebraic manipulation. Here are the steps to solve the equation:

The solutions to the equation will be the values of x that make the equation true.

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Answer:

Step-by-step explanation:

let angle [tex]B[/tex] . be [tex]y[/tex]   then,

using angle sum property,

x+y+90=180 (angle c is 90 because its given)

hence you have the equation

Answer:

Cos() = AC / AB

So in this case: Cos() = 5 / 13

Answer:

Step-by-step explanation:

The most general form of this, without seeing any of the options, would be:

[tex]y=(x+3)^2+6[/tex]

Again, there might have been a value outside the parenthesis that may or may not have been negative, but either way, this is the most basic translation of the parabola.

Answer:

2(3x-2(3x+2)

Step-by-step explanation:

1) 18x^2-8

2) 2(9x^2-4)

3) 2(3x-2)(3x+2)

Answer:

2(3x-2(3x-2) instead of the + 2

Step-by-step explanation:

Answer:

Option 3 and 4

Step-by-step explanation:

We have to find the solutions represented by point A on graph

The point A is represented by (-6,7i)

Lets check all options one by one

1.

[tex]2(1-2i)-(4+3i)\\= (2-4i)-(4+3i)\\=2-4i-4-3i\\=2-4-4i-3i\\=-2-7i[/tex]

This option is not correct

2.

[tex]2(1+2i)+(-4-3i)\\=(2+4i)-4-3i\\=2+4i-4-3i\\=2-4+4i-3i\\=-2+i[/tex]

This option is also not correct

3.

[tex]2(-1+2i)-(4-3i)\\=(-2+4i)-4+3i\\=-2+4i-4+3i\\=-2-4+4i+3i\\=-6+7i[/tex]

This solution is same as point A so this option is correct

4.

[tex](-2+4i)+(-4+3i)\\=-2+4i-4+3i\\=-2-4+4i+3i\\=-6+7i[/tex]

Solution same as point A so this is correct option.

5.

[tex](-2-4i)+(4-3i)\\=-2-4i+4-3i\\=-2+4-4i-3i\\=2-7i[/tex]

Not correct

The correct options are option no 3 and 4 ..

In the complex plane, complex numbers are represented as points with the real part as the x-coordinate and the imaginary part as the y-coordinate. Therefore, various complex number operations like addition, subtraction, multiplication, or division could result in point A, depending on the specific numbers and operations involved.

To understand which operations involving complex numbers would have solutions represented by point A on a graph, we need to firstly understand that in the complex plane, complex numbers are represented as points. The x-axis corresponds to the real part of the number and the y-axis corresponds to the imaginary part. For example, complex number a + bi would correspond to a point at coordinates (a, b) on the graph.

Various operations involving complex numbers could potentially yield the given point A, such as addition, subtraction, multiplication, or division of two complex numbers. However, without the specific coordinates of point A or other pertinent information (such as the specific complex numbers involved in the operation or the nature of the operation), we cannot definitively say which operation would result in point A output.

For instance, addition of two complex numbers can be visually depicted as adding the vectors for the two numbers together. Depending on what the two initial numbers are, their sum could land at point A.

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Answer:

134.38

This value represents 134.38

Step-by-step explanation:

Find the mean.

950+620+545+810+775+1120+905+775=6500

6500/8=812.5

Subtract the mean of numbers.

950-812.5=137.5

812.5-620=192.5

812.5-545=267.5

812.5-810=2.5

812.5-775=37.5

1120-812.5=307.5

905-812.5=92.5

812.5-775=37.5

Find the mean.

137.5+192.5+267.5+2.5+37.5+307.5+92.5+37.5=1075

1075/8≈134.38

This means that 134.38 is $134.38

MAD

To find the MAD of the data, you must first find the mean. To do this, simply add all the numbers up and divide by the number of data values.

950+620+545+810+775+1120+905+775=812.5

So the mean is 812.5 ($812.5).

The final step to finding the MAD is to difference between each data value and the mean.

|950-812.5| + |620-812.5| + |545-812.5| + |810-812.5| + |775-812.5| + |1120-812.5| + |905-812.5| + |775-812.5|

137.5 + 192.5 + 265.5 + 2.5 + 37.5 + 307.5 + 92.5 + 37.5/8

1073/8=9.625

So, 134.125 is the answer.

BUT... the question said round to the nearest cents. So, it's 13413 cents.

Answer:

7/4

Step-by-step explanation:

Answer:

4/7

Step-by-step explanation:

To answer this, take the negative reciprocal of -7/4, obtaining 4/7.  

7-4x= 31

Bring over 7 to the other side

Positive 7 changes into negative 7

7-7-4x= 31-7

-4x= 24

Divide by -4 for -4x and 24

-4x/-4= 24/-4

x= -6

Check answer by using substitution method

7-4x= 31

7-4(-6)= 31

7+24= 31

31= 31

Answer : C. x= -6

Answer:

C. x = -6

Step-by-step explanation:

Isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.

First, subtract 7 from both sides:

7 - 4x = 31

7 (-7) - 4x = 31 (-7)

-4x = 31 - 7

-4x = 24

Divide -4 from both sides to isolate the variable, x:

(-4x)/-4 = (24)/-4

x = 24/-4

x = -6

x = -6, or C. is your answer.

~

Answer:

There are several ways to graph a continuous function. I advise you to graph several points that belongs to the function and then join the points.

If the function is linear, two points will be enough to build the graph. If we have functions of higher degree, is better to plot some points, observe the pattern and then join the points.

If the function is extremly difficult to graph by hand, the help of a graphing calculator may be needed.

To graph a continuous function, identify the independent and dependent variables, ensure the values on axes are evenly spaced, plot values as coordinates, and draw a smooth curve connecting the points without lifting the pen. The area under the graph is significant in calculus for representing continuous change, and for continuous probability distributions, it defines the probability as the area under the curve.

To graph a continuous function, first identify your independent and dependent variables. The independent variable is typically placed on the X-axis, while the dependent variable goes on the Y-axis. Ensure the scales for both axes are evenly spaced to accurately represent the continuous values. After setting up the axes, plot the pairs of values that the function defines as x and y coordinates. For a continuous function, you should be able to draw the curve connecting these points without lifting your pen off the paper.

In the context of calculus, the area under the graph represents the aggregation of continuous change over an interval, which differs from discrete sums used for step changes. This is critical when working with functions to determine areas, volumes, or growth over an interval. As such, if you are graphing a function like f(x) = x², you would plot points for x values from the interval and then draw a smooth curve through those points to represent the function continuously.

When considering continuous probability distributions, the graph will depict a curve known as the probability density function (pdf). Probability is calculated as the area under the curve, emphasizing the importance of a continuous, unbroken line in representing these types of distributions.

Answer:

f(x) = 0.2x - 4  (incorrect)

f(x) = 0.5x + 2  (correct)

f(x) = 1/2x + 2      (correct)

y – 3 = 1/2(x – 2)     (correct)

y – 1 = 0.5(x + 2)

Step-by-step explanation:

Step 1 : Find two coordinates

(0, 2) (-4, 0)

Step 2 : Find the slope

Slope = m = Y2-Y1/X2-X1

m = 0-2/-4-0

m = -2/-4

m = 1/2 or 0.5

Step 3 : Find the y-intercept

Y-intercept is where the line intersects the y-axis

c = 2

Step 4 : Form the equation y=mx + c

Given Equations and their slope intercept forms:

1) f(x) = 0.2x - 4

This is incorrect because slope is 1/2 or 0.5 and y intercept is 2

2) f(x) = 1/2x + 2

y = 1/2x + 2

This is correct because slope is 1/2 or 0.5 and y intercept is 2

3) f(x) = 0.5x + 2

y= 0.5x + 2 (As m=0.5)

This is correct because slope is 1/2 or 0.5 and y intercept is 2

4) y – 3 = 1/2(x – 2)

Rearranging in slope intercept form:

y-3=1/2x - 1

y = 1/2x-1+3

y = 1/2x + 2

This is correct because slope is 1/2 or 0.5 and y intercept is 2

5) y – 1 = 0.5(x + 2)

y -1 = 0.5x+1

y = 0.5x +1+1

y = 0.5x + 2

This is correct because slope is 1/2 or 0.5 and y intercept is 2

!!

first off, let's notice something on this line, the graph touches the y-axis at 2, namely when x = 0, y = 2, so that's the y-intercept for this line.

now, let's notice something else, as the line moves from x = -4, to the right towards x = 0, the run is 4 units, the rise is 2 units, so its slope is rise/run or  2/4 or 1/2, that said, that gives us an equation of

[tex]\bf y=\cfrac{1}{2}x+2\qquad \impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}\implies y=0.5x+2[/tex]

[tex]\bf y-3 = \cfrac{1}{2}(x-2)\implies y-3=\cfrac{1}{2}x-1\implies y=\cfrac{1}{2}x+2\qquad \textit{\Large\checkmark} \\\\\\ y-1=0.5(x+2)\implies y-1=0.5x+1\implies y=0.5x+2\qquad \textit{\Large\checkmark} \\\\\\ f(x) = 0.2x-4\qquad \bigotimes[/tex]

Answer: Choice D

Step-by-step explanation: Why? Because (d-21) is what how many days she was late. The problem is asking what (d-21) presents and Choice D says 'The number of days the book is late'. If the problem did have the 0.10 before (d-21) THEN it would've been Choice C.

In the expression 0.10(d-21), the term (d-21) represents the number of days the book is late, beyond the initial 21-day borrowing period.

The expression (d-21) in the context of the late fee calculation represents the number of days the book is late. Since the library charges a late fee only if the book is returned more than 21 days after it was borrowed, subtracting 21 from the total days borrowed, which is represented by 'd,' gives us the number of days Camilla returned the book late. Multiplying this by the per-day late fee of $0.10 gives us the total late fee owed.

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Answer:

u bought 4 9-volt batteries and

the cost of the 3 AA batteries bought is $4.83

To find out how many 9-volt batteries were bought, you first need to subtract the total cost of AA batteries ($4.83) from the total bill ($14.39) to get the total spent on 9-volt batteries. Then, divide this total by the price of each 9-volt battery ($2.39). The result is 4, so you bought 4 9-volt batteries.

To answer this question, let's first determine how much the total purchase of the AA batteries was. Since AA batteries cost $1.61 each and you bought 3, the total cost for the AA batteries is 3 * $1.61 = $4.83.

Now we need to figure out how much was spent on 9-volt batteries. Subtract the cost of the AA batteries from the total bill: $14.39 - $4.83 = $9.56. This is the total cost of the 9-volt batteries.

The price of each 9-volt battery is $2.39, so to find out how many batteries were bought we divide the total cost of 9-volt batteries by the price of each battery: $9.56 / $2.39 ≈ 4. Therefore, you bought 4 9-volt batteries.

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Answer:

$30 down  

and 10 installments of $10 each

Step-by-step explanation:

paying $30 down means the left price to divide into installments is : 130-30 = 100

since he will pay the rest in equal installments, we divide  by the number of installments to get how much each one will be :

100/10 = $10

Answer:

All 3 angles of any triangle have a sum = 180. Therefore, x + 2x + 3x = 180 6x = 180 x = 180/6 = 30. So It is a 30-60-90 special right triangle!!

Step-by-step explanation:

Answer:

[tex]\large\boxed{x=\dfrac{8}{3}}[/tex]

Step-by-step explanation:

[tex]-3(-x)-6=-3x+10\\\\3x-6=-3x+10\qquad\text{add 6 to both sideS}\\\\3x=-3x+16\qquad\text{add}\ 3x\ \text{to both sides}\\\\6x=16\qquad\text{divide both sides by 6}\\\\x=\dfrac{16}{6}\\\\x=\dfrac{8}{3}[/tex]

Answer:

m=-1

Step-by-step explanation:

9m+13 =4

Subtract 13 from each side

9m+13-13 =4-13

9m = -9

Divide each side by 9

9m/9 = -9/9

m = -1

Answer:

m=-1

Step-by-step explanation:

so we know she has sculptures and paintings, if she sold twice as many paintings as sculptures, that means that for every 2 paintings, she sold 1 sculpture, so the paintings and sculptures are on a 2:1 ratio.

we know she sold a total of 57, so we'll need to split 57 in a 2:1 ratio, we'll simply divide the whole amount of 57 by (2+1) and distribute accordingly.

[tex]\bf \cfrac{paintings}{sculptures}\qquad 2:1\qquad \cfrac{2}{1}\qquad \qquad \cfrac{2\cdot \frac{57}{2+1}}{1\cdot \frac{57}{2+1}}\implies \cfrac{2\cdot 19}{1\cdot 19}\implies \cfrac{\boxed{38}}{19}[/tex]

Answer:  dilation and translation

Step-by-step explanation:

Here Δ ABC and ΔA'B'C' are similar triangles (By AAA similarity postulate), so one of the transformation used here is dilation and also the there is some distance between the triangles without any change in its orientation , so the other transformation used here is translation.

Answer:

Step-by-step explanation:

y = (x^2 + 4x)      + 2

Take 1/2 of the linear term 4/2 = 2 and square that result. 2^2 = 4.

Put it after 4x

y = (x^2 + 4x + 4)   +2  Subtract what you put inside the brackets on the outside.

y = (x^2 + 4x + 4) + 2 - 4      Combine the right.

y = (x^2 + 4x + 4) - 2            Express the brackets as a square.

y = (x + 2)^2 - 2

That's your answer

a = 1

h = 2

k = -2

Answer:

(x + 2)2 − 2

Step-by-step explanation:

Just took the test and got it right

Answer:

The required variable expression is [tex]L= M + 11[/tex]

Step-by-step explanation:

Consider the provided information.

Let m represent the weight of Marcia's dog.

Let L represent the weight of Lisa's dog.

Lisa's dog is 11 pounds heavier than Marcia's dog.

That means if we add 11 pounds in Marcia's dog weight, the weight of Marcia's dog will be equal to Lisa's dog.

This can be written as:

[tex]L= M + 11[/tex]

Where m represent the weight of Marcia's dog and L represent the weight of Lisa's dog.

Hence, the required variable expression is [tex]L= M + 11[/tex]

Answer:

d. no real solutions

Step-by-step explanation:

3p − 9p² = 6

0 = 9p² − 3p + 6

0 = 3p² − p + 2

The discriminant of ax² + bx + c is b² − 4ac.

If the discriminant is negative, there are no real roots.

If the discriminant is zero, there is 1 real root.

If the discriminant is positive, there are 2 real roots.

If the discriminant is a perfect square, the root(s) are rational.

If the discriminant isn't a perfect square, the root(s) are irrational.

Finding the discriminant:

a = 3, b = -1, c = 2

(-1)² − 4(3)(2) = -23

The discriminant is negative, so there are no real roots.

Final answer:

After rewriting the equation 3p-9p²=6 in standard quadratic form and calculating the discriminant, we find that the discriminant is negative, indicating the equation has d. no real solutions.

Explanation:

To determine the number and type of solutions the equation 3p-9p²=6 has using the discriminant, we first need to rewrite the equation in standard quadratic form, which is ax² + bx + c = 0. Moving all terms to one side gives us -9p² + 3p - 6 = 0, where a = -9, b = 3, and c = -6. The discriminant of a quadratic equation is defined as b² - 4ac.

A discriminant greater than zero indicates two real solutions, equal to zero indicates one real solution, and less than zero indicates no real solutions. Calculating the discriminant for our equation: (3)² - 4(-9)(-6)=9-216=-207, which is less than zero. Therefore, the equation -9p²+ 3p - 6 = 0 has d. no real solutions.

Answer:Solution

(

−

4

)

(

7

2

+

3

)

Step-by-step explanation:

7

3

−

2

8

2

+

3

−

1

2

7x^{3}-28x^{2}+3x-12

7x3−28x2+3x−12

Grouping

1

Find one factor

Factor by grouping

Factor by grouping

Factor by grouping

(

−

4

)

(

7

2

+

3

)

{\color{#c92786}{({\color{#c92786}{x-4}})({\color{#c92786}{7x^{2}+3}})}}

(x−4)(7x2+3)

Answer:

m=(-6,6)

Step-by-step explanation:

m^2 -36 = 0

Reorder the terms:

-36 + m^2 = 0

Solving for variable 'm'.

 Add '36' to each side of the equation.

-36 + 36 + m^2 = 0 + 36

Combine like terms: -36 + 36 = 0

0 + m^2 = 0 + 36

m^2 = 0 + 36

Combine like terms: 0 + 36 = 36

m^2 = 36

Simplifying

m^2 = 36

Take the square root of each side:

√m^2=+/-√36

m=(+/-)6

m = {-6, 6}

Answer: About 99.7% of sixth-grade students will have

heights between 51.1 inches and 64.9 inches.

Step-by-step explanation:

According to the empirical rule, 99.7% of data falls within the three standard deviations from the mean.

Given : Mean: [tex]\mu=58\text{ inches}[/tex]

Standard deviation:=  [tex]\sigma=2.3\text{ inches}[/tex]

Then, the  99.7% of sixth-grade students will have  heights between

[tex]\mu-3\sigma[/tex] inches and [tex]\mu+3\sigma[/tex] inches

i.e. [tex]58-3(2.3)[/tex] inches and [tex]58+3(2.3)[/tex] inches

i.e. 51.1 inches and 64.9 inches.

Answer:

About 99.7% of sixth-grade students will have

heights between 51.1 inches and 64.9 inches.

Step-by-step explanation:

Got it right :/

Answer:

(-7,-3)

Step-by-step explanation:

The rule of 90° clockwise rotation is given below:

If a point (x,y) is rotated 90° clockwise, the new point would be (y, -x)

Since the point M is given as (3,-7), the rotated point will be (-7,-3).

Note: you can graph and check also

Answer:

(-7, -3)

Step-by-step explanation:

To find the area, you multiply b×h. (Base × height) b=20 and h=9. Multiply it and you will get 180. The answer is 180 cm².

The area of the parallelogram is equal to 180 square meters.

The space occupied by any two-dimensional figure in a plane is called the area. The space occupied by the parallelogram in a two-dimensional plane is called as the area of the parallelogram.

Given that a parallelogram has a base of 20 cm and a height of 9 cm. The area of the parallelogram is calculated by the formula,

Area of the parallelogram = Base x Height

Area of the parallelogram = 20 x 9

Area of the parallelogram = 180 square meters

Therefore, the area of the parallelogram is equal to 180 square meters.

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Step-by-step explanation:

The slope-intercept form of an equation of a line:

y = mx + b

m - slope

b - y-intercept

y = 1 - 3x = -3x + 1

slope = -3

y-intercept at 1

y = 3x - 1

slope = 3

y-intercept at -1

x - 3 = y → y = 1x - 3

slope = 1

y-intercept at -3

x + 3 = y → y = 1x + 3

slope = 1

y-intercept at 3

Answer:

(X - 4)(x - 4)

Step-by-step explanation:

x2 - 8x + 16

What two numbers multiply to 16 and add to -8

-4 * -4 = 16

-4 +-4 = -8

(x-4) (x-4)

(x-4)^2

Answer:

B

Step-by-step explanation:

(x - 4)(x - 4)

Answer:

8/21

Step-by-step explanation:

(-2/3)/(-7/4)

When dividing, reciprocate the dividend then multiply to the divisor.

Reciprocate

-4/7

Multiply

-2/3 * -4/7

8/21

Answer

8/21

To simplify (-2/3)/(-7/4), we first convert the division operation into multiplication by using the reciprocal of the second fraction. This results in (-2/3)×(-4/7). We, then, multiply the numerators together and the denominators together, leading to the final simplified fraction of 8/21.

To simplify this expression, (-2/3)/(-7/4), it is useful to first understand that division is the same as multiplying by the reciprocal. The reciprocal of a fraction is obtained by switching the numerator and the denominator. The reciprocal of (-7/4) is thus (-4/7). Therefore, the expression becomes (-2/3)×(-4/7).

The next step is to multiply the numerators together and the denominators together. So (-2 × -4) / (3 × 7) gives 8/21. The negative signs cancel each other out as a negative time a negative equals a positive.

So therefore (-2/3)/(-7/4) simplifies to 8/21.

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[tex]\bf \textit{Jim's Gym}\\\\ \begin{array}{cccll} initial~fee&visits&cost\\ \cline{1-3} 300&1&300+3(1)\\ &2&300+3(2)\\ &3&300+3(3)\\ &4&300+3(4)\\ &x&300+3(x) \end{array}\implies y = 300+3x \\\\[-0.35em] ~\dotfill\\\\ \textit{Sally's Salon}\\\\ \begin{array}{cccll} initial~fee&visits&cost\\ \cline{1-3} 250&1&250+5(1)\\ &2&250+5(2)\\ &3&250+5(3)\\ &4&250+5(4)\\ &x&250+5(x) \end{array}\implies y = 250+5x[/tex]

when are the plans equal?

[tex]\bf \stackrel{Jim's}{300+3x}~~=~~\stackrel{Sally's}{250+5x}\implies 50+3x=5x \\\\\\ 50=2x\implies \cfrac{50}{2}=x\implies 25=x[/tex]

So, we can start off by just listing the facts.

The Initial fee for Jim's Gym has an initial fee of $300, and Sally's Salon has an initial fee of $250. Every visit to Jim's Gym costs $3, and Sally's Salon costs $5.

The question is using the variable x, in which x represents the number of visits that person has made.

So the equation for Jim's Gym is 300 + 3x (since, like we've established earlier, it costs $3 per visit.

The equation for Sally's Salon is 250 + 5x (since it costs $5 per visit)

Since we're trying to equalize costs, make the entire equation

300 + 3x = 250 + 5x

Subtract both sides by 250

50 + 3x = 5x

Subtract both sides by 3x

50 = 2x

Divide both sides by 2

x = 25

Option D is the answer.