Find the slope of the line through (3, 7) and (–1, 4)

The augmented matrix for this system is

[tex]\left[\begin{array}{ccc|c}4&-3&1&22\\4&1&5&30\\3&-1&-1&4\end{array}\right][/tex]

Subtract row 1 from row 2, and subtract 3(row 1) from 4(row 3):

[tex]\left[\begin{array}{ccc|c}4&-3&1&22\\0&4&4&8\\0&5&-7&-50\end{array}\right][/tex]

Multiply row 2 by 1/4:

[tex]\left[\begin{array}{ccc|c}4&-3&1&22\\0&1&1&2\\0&5&-7&-50\end{array}\right][/tex]

Subtract 5(row 2) from row 3:

[tex]\left[\begin{array}{ccc|c}4&-3&1&22\\0&1&1&2\\0&0&-12&-60\end{array}\right][/tex]

Multiply row 3 by -1/12:

[tex]\left[\begin{array}{ccc|c}4&-3&1&22\\0&1&1&2\\0&0&1&5\end{array}\right][/tex]

While this isn't exactly RREF, you can already solve the system quite easily:

[tex]\boxed{z=5}[/tex]

[tex]y+z=2\implies\boxed{y=-3}[/tex]

[tex]4x-3y+z=22\implies4x=8\implies\boxed{x=2}[/tex]

We can confirm this solution by continuing with the row reduction. Subtract row 3 from row 2:

[tex]\left[\begin{array}{ccc|c}4&-3&1&22\\0&1&0&-3\\0&0&1&5\end{array}\right][/tex]

Subtract -3(row 2) and row 3 from row 1:

[tex]\left[\begin{array}{ccc|c}4&0&0&8\\0&1&0&-3\\0&0&1&5\end{array}\right][/tex]

Finally, multiply row 1 by 1/4:

[tex]\left[\begin{array}{ccc|c}1&0&0&2\\0&1&0&-3\\0&0&1&5\end{array}\right][/tex]

and we end up with [tex]\boxed{(x,y,z)=(2,-3,5)}[/tex], as before.

Final answer:

To solve for x, y, and z, the given system of equations is represented as an augmented matrix, then reduced to its reduced row-echelon form through row operations, from which the solutions can be directly obtained.

Explanation:

To solve the system of equations by finding the reduced row-echelon form of the augmented matrix, we first write the system as an augmented matrix:

[ 4 -3  1 | 22 ]
[ 4  1  5 | 30 ]
[ 3 -1 -1 |  4 ]
Next, we perform row operations to convert the matrix to its reduced row-echelon form. Once we have the reduced form, we can directly read off the solutions for the variables x, y, and z.

These row operations usually involve scaling rows, adding and subtracting rows, and swapping rows to systematically bring the matrix into the desired form. The final reduced row-echelon form should look like:

[ 1 0 0 | x ]
[ 0 1 0 | y ]
[ 0 0 1 | z ]

Where x, y, and z are the solutions to the original equations. At this stage, each row corresponds to an equation of the form x=..., y=..., z=..., making it straightforward to determine the values of the variables.

Step 1: evaluate f(x+h) and f(x)

We have

[tex]f(x+h) = -(x+h)^2+6(x+h)-5 = -(x^2+2xh+h^2)+6x+6h-5[/tex]

[tex]= -x^2-2xh-h^2+6x+6h-5[/tex]

And, of course,

[tex]f(x)=-x^2+6x-5[/tex]

Step 2: evaluate f(x+h)-f(x)

[tex]f(x+h)-f(x)=-x^2-2xh-h^2+6x+6h-5-(-x^2+6x-5)=-2xh-h^2+6h[/tex]

Step 3: evaluate (f(x+h)-f(x))/h

[tex]\dfrac{f(x+h)-f(x)}{h}=-2x-h+6[/tex]

Step 4: evaluate the limit of step 3 as h->0

[tex]f'(x) = \displaystyle \lim_{h\to 0} \dfrac{f(x+h)-f(x)}{h}=-2x+6[/tex]

So, we have

[tex]f'(1) = -2\cdot 1+6 = 4,\quad f'(2) = -2\cdot 2+6 = 2,\quad f'(3) = -2\cdot 3+6 = 0[/tex]

Answer:$306

Step-by-step explanation:

firstly Mr. Tucker 250 weekly

sold 2800 appliances and earn 2%, so find the 2% of 2800 which is

x/2800 X 2/100 = 56

this mean he earn $56 dollars on the sales . add his weekly earn which is $250 to the $56  which will be $250 + 56 = $306 for the week

Answer:

306$

Step-by-step explanation:

2,800*0.02=56

250+56=306

Answer:

B. 164°

Step-by-step explanation:

arc JL = 2 (<JKL)

arc JL = 2(82)

arc JL = 164°

The measure of JL is 164°.

The correct option is (B)

An arc whose measure is less than 180 degrees is called a minor arc.

Given: angle JKL= 82°

We know by the theorem that

"When two angles are subtended by the same arc, the angle at the centre of a circle is twice the angle at the circumference."

Then,

JL= 2 (JKL)

JL= 2(82)

JL= 164°.

Hence, the measure of JL is 164°.

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Answer:

30(3+2)

Step-by-step explanation:

90=3(30)=3(3)(10)=3(3)(2)(5)

60=3(20)=3(5)(4)=3(2)(2)(5)

The factors that 90 and 60 have in common are a pair of 3,2, and 5's.

So the biggest factor we can factor out is 3*2*5 which is 30

So 30(3+2)

Leftovers from the prime factorizations above stayed in the ( )

Answer:

x= 6.5 cm

Step-by-step explanation:

When a tangent line touches the circle, it forms a right angle triangle at that point

Apply the Pythagorean relationship in this case

Given that the height is = 20.2 cm = b

The hypotenuse is = c= x+14.7 cm

General formulae is;

a² +b² =c²

x² + 20.2² =( x+ 14.7)²

x² + 408.04= x² +14.7x+14.7x+216.09

x² + 408.04= x² + 29.4 x +216.09.........................collect like terms

x²-x² + 408.04-216.09= 29.4x

191.95= 29.4x-------------------------------divide by 29.4 t0 get x

191.95/29.4 =x

x=6.5 cm

Answer: Options 'A', 'C' and 'F' are correct.

Step-by-step explanation:

Since we have given that

Corresponding angles are those angles which takes the same corresponding position at intersection when a transversal cut the two parallel lines.

so, According to this , we get that

∠1 and ∠5

∠2 and ∠6

∠3 and ∠7

∠4 and ∠8

so, Options 'A', 'C' and 'F' are correct.

Answer:

3x - 2y = 10

Step-by-step explanation:

We are given that the sum of two numbers is 0 and twice the smaller number subtracted from 3 times the larger number is 10.

Assuming x to be the large number and y to be the smaller number we can write an equation to represent this.

Sum of two numbers is 0:

[tex]x+y=0[/tex]

Twice the smaller number subtracted from 3 times the larger number is 10:

[tex]3x-2y=10[/tex]

Answer:

a) translation of 1 unit up

Step-by-step explanation:

I was expecting a choice that said A(1,4) is in the first quadrant so 90 degrees clockwise is fourth quadrant.  For perpendicularity we reverse the coordinates, negating one of them.  For the fourth quadrant, it must be the y coordinate that's negative.  We end up at A'(4,-1).

The answer is the second choice:  create a circle with the center at the origin.  The image of A' will be on the circle, 90 degrees clockwise from A.

Answer:

200

Step-by-step explanation:

150=75% * X

X= 150/0.75

X-200

Darry is 200 centimeters tall

These are the common measurements:

Given,

Calvin = 150 centimeters and 75% of Darryl's height.

we have to find x

150=75% * X

X= 150/0.75

X-200

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Answer:

The answer is 2

Step-by-step explanation:

Answer: The slope of C'D' = 2

Step-by-step explanation:

From the given picture, we can that the coordinates of point C and D are (5,4) and (4,2).

After dilation with scale factor of 8 with the origin as the center of dilation , the coordinates of C' and D' will be :-

[tex]C'=(8\times5,8\times4)=(40,32)[/tex]

[tex]D'=(8\times4,8\times2)=(32,16)[/tex]

Now, the slope of line segment C'D' will be

[tex]\text{Slope}=\dfrac{\text{Change in y-coordinate}}{\text{Changein x-coordinate}}\\\\\Rightarrow\text{Slope}=\dfrac{16-32}{32-40}\\\\\Rightarrow\text{Slope}=\dfrac{-16}{-8}\\\\\Rightarrow\text{Slope}=2[/tex]

Answer: C

Step-by-step explanation:

Final answer:

The correct symbol to complete the inequality 18.45_3+h is ≤, representing that Leticia spends at most $3 more than Humberto, so her spending of $18.45 is less than or equal to Humberto's spending plus $3.

Explanation:

Leticia spends a certain amount on a shirt, and we have an inequality to show that she spends at most $3 more than Humberto. The inequality to represent this situation, where h represents the amount Humberto spends, is 18.45 ≤ 3 + h. This means that Leticia's spending of $18.45 is less than or equal to Humberto's spending plus an additional $3. The symbol ≤ (less than or equal to) completes the inequality because it indicates that the amount Leticia spends is not greater than the cost of the shirt plus an extra $3. In other words, Humberto's spending could be exactly $3 less than Leticia's, or even less, but not more.

Answer:

C

Step-by-step explanation:

Under a counterclockwise rotation about the origin of 180°

a point (x, y ) → (- x, - y) → C

Answer:  C) (x, y) → (−x, −y)

Step-by-step explanation:

When we rotate a figure 180°clockwise or counter-clockwise , the magnitude of x and y coordinates remains same but their signs got changed.

For example : After a rotation of 180°clockwise or counter-clockwise (3,4) becomes (-3,-4).

Algebraically , we can say

Ordered pair (x , y ) will become (-x, -y) after a rotation of 180°clockwise or counter-clockwise.

Thus , the algebraic rule describes the 180° counter-clockwise rotation about the origin will be :-

C) (x, y) → (−x, −y)

Answer:

cos 2Ф = - 161/289 , tan 2Ф = - 240/161

Step-by-step explanation:

* Lets explain how to solve the problem

∵ cos Ф = - 8/17

∵ Ф lies in the 3rd quadrant

- In the 3rd quadrant sin and cos are negative values, but tan is

 a positive value

∵ sin²Ф + cos²Ф = 1

∴ sin²Ф + (-8/17)² = 1

∴ sin²Ф + 64/289 = 1

- Subtract 64/289 from both sides

∴ sin²Ф = 225/289 ⇒ take √ for both sides

∴ sin Ф = ± 15/17

∵ Ф lies in the 3rd quadrant

∴ sin Ф = -15/17

∵ cos 2Ф = 2cos²Ф - 1 ⇒ the rule of the double angle

∵ cos Ф = - 8/17

∴ cos 2Ф = 2(-8/17)² - 1 = (128/289) - 1 = - 161/289

* cos 2Ф = - 161/289

∵ tan 2Ф = sin 2Ф/cos 2Ф

∵ sin 2Ф = 2 sin Ф × cos Ф

∵ sin Ф = - 15/17 and cos Ф = - 8/17

∴ sin 2Ф = 2 × (-15/17) × (-8/17) = 240/289

∵ cos 2Ф = - 161/289

∴ tan 2Ф = (240/289)/(-161/289) = - 240/161

* tan 2Ф = - 240/161

Answer:

so look the answer is 2090909876

Step-by-step explanation:

Answer:

[tex]P =\frac{1}{12}[/tex]

Step-by-step explanation:

The probability is defined as the number of ways to obtain the desired result among the number of possible outcomes.

The number of possible ways to select 3 hatfields from a group of 5 hatfields is:

[tex]3C5 =\frac{5!}{3!(5-3)!} =10[/tex]

The number of ways to select 3 people from a group of 10 is:

[tex]10C3 =\frac{10!}{3!(10-3)!} =120[/tex]

Then the probability is:

[tex]P =\frac{10}{120}[/tex]

[tex]P =\frac{1}{12}[/tex]

well, the recursive rule of aₙ = aₙ₊₁ + 7, where a₁ = 15, is simply saying that

we start of at 15, and the next term is obtained by simply adding 7, and so on.

well, that's the recursive rule.

so then let's use that common difference and first term for the explicit rule.

[tex]\bf n^{th}\textit{ term of an arithmetic sequence} \\\\ a_n=a_1+(n-1)d\qquad \begin{cases} a_n=n^{th}\ term\\ n=\textit{term position}\\ a_1=\textit{first term}\\ d=\textit{common difference}\\ \cline{1-1} a_1=15\\ d=7 \end{cases} \\\\\\ a_n=15+(n-1)7\implies a_n=15+7n-7\implies a_n=7n+8[/tex]

Subtract the new amount from the original amount:

64 - 85 = -21

Now divide that by the original amount:

-21 / 85 = -0.247

Multiply that by 100 for the percentage:

-0.247 x 100 = -24.7%

Rounded to the nearest percent is -25%

Answer:

3(x2 + 1) + 2

A. (3x + 2)(x2 + 1) WRONG bc = 3x^3

B. 3x2 + 1 + 2 Wrong bc 3x^+3

C. (3x + 2)2 + 1 wrong bc 6x+5

D. 3(x2 + 1) + 2Correct

The problem can be solved using trigonometric principles where the hypotenuse measures 16 inches.

The problem involving a triangular brace can be solved using trigonometry principles. As it involves a right triangle, you can use the sine function to solve the problem. In a right triangle, the sine of an angle (hypotenuse) is the length of the side opposite the angle divided by the length of the hypotenuse.

In this case, you have the opposite side (8 inches), the angle (30 degrees), and the adjacent side (base, 11 inches). Using these values and the sine rule, we can determine the length of the hypotenuse. In a 30-degree angle, the sine is 0.5, so the unknown length can be obtained by dividing the length of the opposite side by the sine of the angle. This gives: 8/0.5 = 16 inches.

Therefore, the correct statement will be that the hypotenuse of the triangular brace is 16 inches.

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Using the principles of trigonometry, specifically the sin and cos functions, applied to a triangle with a 30 degrees angle and known lengths of the sides opposite and adjacent to this angle, we can calculate the length of the hypotenuse.

The question asked pertains to the principles of trigonometry. In particular, we have a triangle with one angle of 30 degrees, and we know the lengths of the side opposite this angle (8 inches) and the base of the triangle, which is adjacent to this angle (11 inches).

With a 30 degree angle, we can use sin, cos, and tan functions to form relationships with the opposite, adjacent, and hypotenuse sides of the triangle. In this scenario, the sin(30 degrees) = opposite length/hypotenuse = 8 inches/hypotenuse. Alternatively, the cos(30 degrees) = adjacent length/hypotenuse = 11 inches/hypotenuse.

To solve for the hypotenuse using the sin, you'd do 8/sin(30 degrees). From the cos, you'd do 11/cos(30 degrees).

Note that while you have two ways to solve for the hypotenuse, these should give you the same answer if your angle and side lengths are correct.

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Answer:

x=(c-b)/3

Step-by-step explanation:

we have

3x+b=c

Solve for x

That means -----> Clear variable x

Subtract b both sides

3x+b-b=c-b

3x=c-b

Divide by 3 both sides

x=(c-b)/3

Answer:

27.6 degrees

Step-by-step explanation:

Use Cosine rule on Acute triangle

b² = a² + c² - 2ac Cos B

where b = 10, a = 14, c = 20

10² = 14² + 20² - 2(14)(20) Cos B

-496 = -560 Cos B

Cos B = (-496) / (-560)

B = [tex]Cos^{-1}[/tex] (-496) / (-560) = 0.483 radians = 27.6 degrees

Answer:

5

Step-by-step explanation:

85 * 18 / 18 * 17

18 cancels out.

85/17

= 5

Your answer for this question is 5

1.) When you divide 85/18 by 17/18, the first step you want to take is switch the reciprocal.

[tex]\frac{85}{18}[/tex]÷[tex]\frac{17}{18}[/tex]=

[tex]\frac{85}{18}[/tex]×[tex]\frac{18}{17}[/tex]

2.) When you switch the reciprocal, you then division sign is changed to times.

[tex]\frac{85}{18}[/tex]×[tex]\frac{18}{17}[/tex]

3.) Now you simply multiply

[tex]\frac{85}{18}[/tex]×[tex]\frac{18}{17}[/tex]=[tex]\frac{1530}{306}[/tex]

4.) Last step, you divide

[tex]\frac{1530}{306}[/tex] = 5

Hope This Helps.

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Answer:

[tex]\sqrt{7(-14+i)}[/tex]

Step-by-step explanation:

We need to find the [tex]\sqrt{-98+7i}[/tex]

We know that 98/7 = 14.

Taking 7 common from both terms we get

[tex]\sqrt{7(-14+i)}[/tex]

Since 7 and -14 are not the perfect squares, so our answer is:

[tex]\sqrt{7(-14+i)}[/tex]

Answer:

Here is your answer in the picture..

Answer:

x = 5

Step-by-step explanation:

Given

9x + 2 = 5x + 22 ( subtract 5x from both sides )

4x + 2 = 22 ( subtract 2 from both sides )

4x = 20 ( divide both sides by 4 )

x = 5

Step-by-step answer:

Area of a kite is half of the product of the diagonals.

The length of diagonal in the x-direction is 4+5 = 9

The length of diagonal in the y-direction is 4+4 = 8

Therefore

Area of kite = 8*9/2 = 36 units.

ANSWER

The correct answer is A.

EXPLANATION

If you know the diagonals of a kite you can easily find the area.

The area of a kite is half the product of the diagonals.

From the graph, the from -5 to 4.

Using the number line approach. The longer diagonal is

[tex] |4 - - 5| = |4 + 5| = |9| = 9 \: \: units[/tex]

Similarly the shorter diagonal is from -4 to 4

[tex] |4 - - 4| = |4 + 4| = |8| = 8 \: \: units[/tex]

The area of the kite is:

[tex]Area= \frac{1}{2} \times 8 \times 9[/tex]

[tex]Area=4 \times 9[/tex]

This implies that

[tex]Area=36 \: square \: \: units[/tex]

The first choice is correct.

Answer:

x=44

Step-by-step explanation:

Making the equation in terms of x:

0.2x + 0.8 = 9.6

-0.8              -0.8

0.2x=8.8

*5      *5

x=44

Answer:

x=44

Step-by-step explanation:

Multiply by 10 from both sides of equation.

0.2x*10+0.8*10=9.6*10

Simplify.

2x+8=96

Subtract by 8 from both sides of equation.

2x+8-8=96-8

Simplify.

96-8=88

2x=88

Divide by 2 from both sides of equation.

2x/2=88/2

Simplify, to find the answer.

88/2=44

x=44 is the correct answer.

I hope this helps you, and have a wonderful day!

Answer:

C, D and E

Step-by-step explanation:

The denominator of f(x) cannot be zero as this would make f(x) undefined.

Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non zero for these values then they are vertical asymptotes.

solve 3x(x - 1)(x + 5) = 0

Equate each factor to zero and solve for x

3x = 0 ⇒ x = 0

x - 1 = 0 ⇒ x = 1

x + 5 = 0 ⇒ x = - 5

Vertical asymptotes at x = -5, x = 1 and x = 0

Answer:

0, 1, -5

Step-by-step explanation:

[tex]\bf 5x-2y<-8\implies -2y<-5x-8\implies \stackrel{\textit{multiplication by a negative}}{y~~\stackrel{\downarrow }{>}~~\cfrac{-5x-8}{-2}} \\\\\\ y>\cfrac{5x+8}{2}\implies y>\cfrac{5}{2}x+4\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}[/tex]

[tex]3x^2-10x=-2\\3x^2-10x+2=0\\\\\Delta=(-10)^2-4\cdot3\cdot2=100-24=76[/tex]

Answer:

The discriminate is 76

Step-by-step explanation:

* Lets explain what is the discriminant

- In the quadratic equation ax² + bx + c = 0, the roots of the

 equation has three cases:

1- Two different real roots

2- One real root or two equal real roots

3- No real roots means imaginary roots

- All of these cases depend on the discriminate value (D)

- The discriminate D = b² – 4ac determined from the coefficients of  

  the equation ax² + bx + c = 0.

# If the value of D positive means greater than 0

∴ There are two different real roots

# If the value of D = 0

∴ There are two equal real roots means one real root

# If the value of D is negative means smaller than 0

∴ There is real roots but the roots will be imaginary roots

∴ We use the discriminant to describe the roots

* Lets solve the problem

∵ 3x² - 10x = -2

- Put it in the form of ax² + bx + c = 0

- Add 2 for both sides

∴ 3x² - 10x + 2 = 0

- Compare between this equation and the form up to find a , b , c

∵ 3x² - 10x + 2 = 0 and ax² + bx + c = 0

∴ a = 3 , b = -10 , c = 2

- Lets find the discriminate D

∵ D = b² - 4ac

∵ a = 3 , b = -10 , c = 2

∴ D = (-10)² - 4(3)(2)  

∴ D = 100 - 24 = 76

* The discriminate is 76