3. Find the point of intersection of the graphs for each system. a) x - y = 1 and 3x - y = -1​

Answer:

Answer for (a)   -2x^4/3

Answer for (b)  2^5x/4^3x

Answer:

a. [tex]-2x\sqrt[3]{x}[/tex]

b. [tex]\frac{1}{2^x}[/tex]

Step-by-step explanation:

a.

Original equation:

[tex]\frac{(-x)^4}{4x}*\frac{8(-x)^{-3}}{x^{-\frac{4}{3}}}[/tex]

So (-x)^4 can be seen as (-x * -x) * (-x * -x), which becomes x^2 * x^2 = x^4, the negatives cancel out of the degree is even. So it becomes:

[tex]\frac{x^4}{4x}*\frac{8(-x)^{-3}}{x^{-\frac{4}{3}}}[/tex]

Cancel out one of the x's on the left fraction:

[tex]\frac{x^3}{4}*\frac{8(-x)^{-3}}{x^{-\frac{4}{3}}}[/tex]

Rewrite the exponent in the numerator: [tex]a^{-x} = \frac{1}{a^x}[/tex]

[tex]\frac{x^3}{4}*\frac{8*\frac{1}{-x^3}}{x^{-\frac{4}{3}}}[/tex]

Simplify the numerator:

[tex]\frac{x^3}{4}*\frac{\frac{8}{-x^3}}{x^{-\frac{4}{3}}}[/tex]

Keep numerator, change division to multiplication, flip the denominator:

[tex]\frac{x^3}{4}*\frac{8}{-x^3} * \frac{1}{x^{-\frac{4}{3}}}[/tex]

multiply the denominator using the exponent identity: [tex]x^a*x^b=x^{a+b}[/tex]

[tex]\frac{x^3}{4}*\frac{8}{-x^{\frac{5}{3}}}[/tex]

Multiply the numerators and denominators:

[tex]\frac{8x^3}{-4x^{\frac{5}{3}}}[/tex]

Use the fact that: [tex]\frac{x^a}{x^b}=x^{a-b}[/tex] to divide the x^3 and x^(5/3) and divide the 4 by the -8

[tex]-2x^{\frac{4}{3}}[/tex]

Rewrite the exponent using the exponent identity: [tex]x^{\frac{a}{b}} = \sqrt[b]{x^a}=\sqrt[b]{x}^a[/tex]

[tex]-2\sqrt[3]{x^4}[/tex]

Rewrite as two radicals: [tex]\sqrt[n]{a} * \sqrt[n]{b} = \sqrt[n]{ab}[/tex]

[tex]-2\sqrt[3]{x^3} * \sqrt[3]{x}[/tex]

Simplify:

[tex]-2x\sqrt[3]{x}[/tex]

b.

[tex]2^{2x}\div4^{3x}*64^{\frac{x}{2}}[/tex]

Rewrite the 4 as 2^2

[tex]2^{2x}\div(2^2)^{3x}*64^{\frac{x}{2}}[/tex]

Use the exponent identity: [tex](x^a)^b=x^{ab}[/tex]

[tex]2^{2x}\div2^{6x}}*64^{\frac{x}{2}}[/tex]

Use the exponent identity: [tex]\frac{x^a}{x^b}=x^{a-b}[/tex]

[tex]2^{2x-6x} = 2^{-4x}[/tex]

Rewrite this part using the definition of a negative exponent: [tex](\frac{a}{b})^{-x} = \frac{b}{a^x}[/tex].

[tex]\frac{1}{2^{4x}} * 64^{\frac{x}{2}}[/tex]

Multiply:

[tex]\frac{64^{\frac{x}{2}}}{2^{4x}}[/tex]

rewrite 64 as 2^6

[tex]\frac{(2^6)^{\frac{x}{2}}}{2^{4x}}[/tex]

Use the identity: [tex](x^a)^b=x^{ab}[/tex]

[tex]\frac{2^{3x}}{2^{4x}}[/tex]

Use the identity: [tex]\frac{x^a}{x^b}=x^{a-b}[/tex]

[tex]2^{-x}[/tex]

rewrite using the definition of a negative exponent: [tex](\frac{a}{b})^{-x} = \frac{b}{a^x}[/tex]

[tex]\frac{1}{2^x}[/tex]

Answer:

Let the no. be x

51 + x = -1

x = -1-51

x = 1 + 51

Answer:

-52

Step-by-step explanation:

51 + (-52) = -1

Answer: 34

Step-by-step explanation:

[tex]V=\frac{4}{3}(\pi)(2^{3}) \approx \boxed{34}[/tex]

The complete equation of the polynomial is 2x(x^2 - 11x) + 10x^3 = 3x(4x^2 - 7x) - x^2

The equation is given as:

_x(x^2 - _x) + _x^3 = _x(_x^2 + _x) - x^2

Complete the blanks using alphabets

ax(x^2 - bx) + cx^3 = dx(ex^2 + fx) - x^2

Open the brackets

ax^3 - abx^2 + cx^3 = dex^3 + dfx^2- x^2

Factorize the expression

(a + c)x^3 - abx^2 = dex^3 + (df - 1)x^2

By comparison, we have:

a + c = de

-ab = df - 1

Rewrite the second equation as:

ab + df = 1

So, we have:

a + c = de

ab + df = 1

Set a = 2 and c = 10.

So, we have:

a + c = de ⇒ de = 2 + 10 ⇒ de = 12

ab + df = 1 ⇒2b + df = 1

Express 12 as 3 * 4 in de = 12

de = 3 * 4

By comparison, we have:

d = 3 and e = 4

So, we have:

2b + df = 1

This gives

2b + 3f = 1

Set b = 11.

So, we have:

2 * 11 + 3f = 1

This gives

22 + 3f = 1

Subtract 22 from both sides

3f = -21

Divide by 3

f = -7

Hence, the complete equation is:

2x(x^2 - 11x) + 10x^3 = 3x(4x^2 - 7x) - x^2

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Answer:

Proof below

Step-by-step explanation:

A number, p, is divisible by another number, d, if and only if there is some non-negative integer, n, such that n*d=p.

To prove that, 300 is divisible by 10 because, 30 is a non-negative integer, and 10*30=300.

Strategies for Divisibility by a composite number

Note that in the previous example, 10 is a composite number.  This means that both one 2 and one 5 (the full list of 10s factors) had to be factored out of the 300.

In the given problem, we are to prove that the number is divisible by 14.  Observe 14 is composite with factors of 2 and 7.

Since the expression is given with exponents, it will be helpful to recall a few exponent properties to algebraically manipulate the expression.

Recall the following property of exponents:

Original expression...

[tex]8^{10}-2^{27}[/tex]

Recognizing 8 as a power of 2...

[tex](2^3)^{10}-2^{27}[/tex]

Simplifying and rewriting so that both terms are powers of 2...

[tex]2^{30}-2^{27}[/tex]

Observing that both terms have 27 twos as factors...

[tex]2^{27}*2^{3}-2^{27}[/tex]

Factoring out 27 twos...

[tex]2^{27}*(2^{3}-1)[/tex]

Simplifying the expression in the parenthesis:

[tex]2^{27}*(8-1)[/tex]

[tex]2^{27}*(7)[/tex]

Knowing that we also need a factor of 2, use properties of exponents, and associative property of multiplication...

[tex](2^{26}*2^1)*7[/tex]

[tex]2^{26}*(2^1*7)[/tex]

[tex]2^{26}*(2*7)[/tex]

[tex]2^{26}*14[/tex]

Since 2^26 is a non-negative integer, the original expression is divisible by 14.

Answer:

$17578.13

Step-by-step explanation:

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Answer: x = 32

Step-by-step explanation:

If m∠ORS = m∠SRQ = 3x - 6 and since the two angles are a linear pair they total 180°. This means half or only one section = 90°

solve for x

3x - 6 = 90

add 6 to both sides

3x = 96

divide each side by 3

x = 32

The graph which represents the relationship between the year and the computers value is; Choice A.

According to the task content, it follows that the computer loses 1/3 of its value every year. Hence, it follows that the value of the computer each year is 2/3 of its previous value.

Hence, the relationship can best be modelled by an exponential function, a decay function to be more specific and can best be represented by Choice A.

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Hello,

2³ = 2 × 2 × 2 → answer A

Step-by-step explanation:

1.) ˆ B = 7∘ ,  ˆ A = 28 ∘ ,  a = 27

-----------------------------------------------------------------------------------------

2.) c ≈ 28.0

-----------------------------------------------------------------------------------------

3.) The triangle has the following measures.

A  =  36 ˚

a  =  14  units

B  =  105 ˚  

b  =  23 units

C  =  39 ˚  

c  = 15  units

------------------------------------------------------------------------------------------

4.) a  ≅  15.0 units

B  =  145 ˚

b  ≅  22.9 units

The distance from the center of dilation, P, to.the image vertice S' is; 6 units.

It follows from the task content that the center of dilation of the triangle QRS is point P and the length of segment PS in the pre-image is; 8 units.

Hence, since the dilation factor as given in the task content is; three-fourths, it therefore follows that the distance of point P to S' in the image is; (3/4) × 8 = 6units.

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Step-by-step explanation:

a11 = 4 × a2

a + 10d = 4 × (a + d)

a + 10d - 4d = 4a

a + 6d = 4a

6d = 4a - a

6d = 3a

(6/3)d = a

2d = a ----- eq. 1

now, sum of 7 terms = s7 = 175

7/2 {2a+6d} = 175

7/2 { (2 × 2d ) + 6d} = 175

7 ( 4d + 6d ) = 175 × 2

7 ( 10d ) = 350

70d = 350

d = 350/70

d = 5

Now from eq. 1,

2d = a

2(5) = a

a = 10, d = 5

THANK YOU........

The difference for both the rhombus and square are as follows;

Therefore, the difference for both the shapes(square and rhombus)are as follows:

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Answer:

m = 2

Step-by-step explanation:

[tex]m = \frac{rise}{run} =\frac{y_{2}-y_1}{x_2-x_1}[/tex]

[tex]=\frac{6 - 8}{2 - 3} \\= \frac{-2}{-1}\\ = 2[/tex]

[tex]\huge\text{Hey there!}[/tex]

[tex]\huge\textbf{The formula for finding the slope:}[/tex]

[tex]\mathbf{\dfrac{y_2 - y_1}{x_2 - x_1} = slope}[/tex]

[tex]\huge\textbf{Some people understand formula like:}[/tex]

[tex]\mathbf{\dfrac{rise}{run} = slope}[/tex]

[tex]\huge\textbf{Breaking down the question:}[/tex]

[tex]\mathbf{y_2 \rightarrow \boxed{\textbf{6}}}[/tex]

[tex]\mathbf{y_1 \rightarrow \boxed{\textbf{8}}}[/tex]

[tex]\mathbf{x_2 \rightarrow \boxed{\textbf{2}}}[/tex]

[tex]\mathbf{x_1 \rightarrow \boxed{\textbf{3}}}[/tex]

[tex]\huge\textbf{What your equation should look like:}[/tex]

[tex]\mathbf{\dfrac{6 - 8}{2 - 3} = slope}[/tex]

[tex]\huge\textbf{Simplify the equation we came up with:}[/tex]

[tex]\mathbf{\dfrac{6 - 8}{2 - 3} = slope}[/tex]

[tex]\mathbf{\rightarrow \dfrac{-2}{-1} = slope}[/tex]

[tex]\mathbf{\rightarrow -2 \div -1 = slope}[/tex]

[tex]\mathbf{\rightarrow 2 = slope}[/tex]

[tex]\huge\textbf{Thus, the answer should most likely be:}[/tex]

[tex]\huge\boxed{\mathsf{Slope \rightarrow \boxed{\textsf{2}}}}}\huge\checkmark[/tex]

[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]

Answer:

y = - 6/7 x - 10/7

Step-by-step explanation:

Slope = (y2-y1) / (x2-x1)

= 6/-7 = -6/7

y = -6/7 x - a

passes by (-4,2)

2 = 24/7 - a, a = -10/7

y = -6/7 x - 10/7

The perimeter of each region is 12 + 1.5π if a circle with an area of 36π is divided into 8 congruent regions

An equation is an expression that shows the relationship between two or more numbers and variables.

Let r represent the area of the circle, hence:

Area = πr²

36π = πr²

r = 6 units

Perimeter = 2πr = 2π(6) = 12π

The circle has 8 congruent regions, hence:

Perimeter of each region = 6 + 6 + 12π/8 = 12 + 1.5π

The perimeter of each region is 12 + 1.5π if a circle with an area of 36π is divided into 8 congruent regions

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The quotient of the complex numbers is 3[cos(240)+ isin(240)] option (D) is correct.

It is defined as the number which can be written as x+iy where x is the real number or real part of the complex number and y is the imaginary part of the complex number and i is the iota which is nothing but a square root of -1.

We have two complex number is given.

The quotient of the complex numbers:

[tex]= \rm \dfrac{12\left[cos\left(320\right)+isin\left(320\right)\right]}{4\left[cos\left(80\right)+isin\left(80\right)\right]}[/tex]

= 3[cos(320-80)+ isin(320-80)]

= 3[cos(240)+ isin(240)]

Thus, the quotient of the complex numbers is 3[cos(240)+ isin(240)] option (D) is correct.

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We have proven that BC = DE using the Segment addition postulate

The segment addition postulate states that given two points A and C, a third point B lies on the line segment AC if and only if the distances between the points satisfy the equation AB + BC = AC

The reasons for the proof are given below:

Statement                                Reason

BD = BC + CD                          Segment addition postulate

CE = CD + DE                           Segment addition postulate

BD = CE                                    Given

BC + CD = CD + DE                  Substitution property

BC = DE                                   Subtraction property

Hence, we have proven that BC = DE using the Segment addition postulate

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The average speed in approximately 184.62 miles per hour.

We have been given that in the race Math 600, drivers must complete 200 laps of 3 miles each. Dave drove the first 150 laps at 180 mph and the last 50 laps at 200 mph.

First of all, we will find the number of miles in 150 laps and 50 laps as:

150 X 3 miles = 450 miles

50 X 3 miles = 150 miles.

Now, we will find time taken to complete each distance as:

Time = 450/180 =2.5 hr

Time = 150/200 = 0.75 hr

Average speed = 184.62

The average speed in approximately 184.62 miles per hour.

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Answer: C. He can show that the measures of arcs AB, AC, and BC are all equal to 120°.

Step-by-step explanation:

Each angle of an equilateral triangle measures 60°, so by the inscribed angle theorem, each arc created by a pair of adjacent vertices will measure 120°.

Answer:

25

Step-by-step explanation:

sin © =36/(35+x)=21/35

36/(35+x)=21/35

x=(36*35/21)-35

x=25

Step-by-step explanation:

SOLUTION::

please watch the image for explanation

Answer:

y = 5

Step-by-step explanation:

Step-by-step explanation:

the slope of a line is the ratio "y coordinate change / x coordinate change" when going from one point on the line to another.

when the slope is 0, it means the y coordinate change is 0 for all points. that means y is a constant. and the line is a horizontal line.

to go through (-2, 5) this constant y has to be the y value of the point.

so,

y = 5

that's it. that is the whole equation for that line, because the slope m is 0. what remains is the y-axis intercept (y = 5).

When the above complex expression has been simplified the solution to the above expression is = 2003/8016.

The formula is given as:

1 - [tex]\frac{1}{n}[/tex] = [tex]\frac{n-1}{n}[/tex]

From the stated express, we have:

(1 - (1/2)) * (1 - (1/3)) * (1 - (1/4)) *( 1 - (1/2004))

→ ((2-1)/2)  ((3-1)/3) ((4-1)/4) ((2004 -1)/2004)

→ (1/2) * (2/3) * (3/4) * (2003/2004)

→ [tex]\frac{ 1* 2003}{4*2004}[/tex]

= 2003/8016

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Answer:

When the above complex expression has been simplified the solution to the above expression is = 2003/8016.

What is the formula for solving the above?

The formula is given as:

1 -  =

From the stated express, we have:

(1 - (1/2)) * (1 - (1/3)) * (1 - (1/4)) *( 1 - (1/2004))

→ ((2-1)/2)  ((3-1)/3) ((4-1)/4) ((2004 -1)/2004)

→ (1/2) * (2/3) * (3/4) * (2003/2004)

→

= 2003/8016

Step-by-step explanation:

Answer:

[tex]\frac{1}{8}[/tex]

Step-by-step explanation:

2 ( [tex]\frac{1}{4}[/tex] )²

= 2 × [tex]\frac{1}{16}[/tex] ← cancel 2 and 16 by 2

= 1 × [tex]\frac{1}{8}[/tex]

= [tex]\frac{1}{8}[/tex]

There are 15 rows of seats in the theater

The given parameters are:

The number of rows is calculated as:

Rows = Total number of people/People per row

This gives

Rows = 357/24

Evaluate the quotient

Rows = 14.875

Round up the number

Rows = 15

Hence, there are 15 rows of seats in the theater

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Answer:

C) 12 m

Step-by-step explanation:

Answer:

680

Step-by-step explanation:

14(-6x +10) - 16x - 100x + 140 for x = -2.

14(-6(-2) + 10) - 16(-2) - 100(-2) + 140

14(12 + 10) + 32 + 200 + 140

14(22) + 372

308 + 372

680

Answer:

The solution of the given set in interval form is [tex]$(-\infty,-4] \cup[12, \infty)$[/tex].

Step-by-step explanation:

It is given in the question an inequality as [tex]$|x-4| \geq 8$[/tex].

It is required to determine the solution of the inequality.

To determine the solution of the inequality, solve the inequality [tex]$x-4 \geq 8$[/tex] and, [tex]$x-4 \leq-8$[/tex]

Step 1 of 2

Solve the inequality [tex]$x-4 \geq 8$[/tex]

[tex]$$\begin{aligned}&x-4 \geq 8 \\&x-4+4 \geq 8+4 \\&x \geq 12\end{aligned}$$[/tex]

Solve the inequality [tex]$x-4 \leq-8$[/tex].

[tex]$$\begin{aligned}&x-4 \leq-8 \\&x-4+4 \leq-8+4 \\&x \leq-4\end{aligned}$$[/tex]

Step 2 of 2

The common solution from the above two solutions is x less than -4 and [tex]$x \geq 12$[/tex].

The solution set in terms of interval is [tex]$(-\infty,-4] \cup[12, \infty)$[/tex].

Answer:

The function f has an inverse because it is one-to-one.

Step-by-step explanation:

The condition for a function to have an inverse is that every x coordinate maps to a single y - coordinate. As illustrated by the diagram attached in the second picture, every x value maps to one y value therefore satisfying the condition making the function have an inverse.

The petrol cost for Anne's journey of 120km is $16.236

Her car has a fuel consumption of 8.2 litres per 100 km .

Therefore,

100 km = 8.2 litres

120 km = ?

cross multiply

volume(litres) = 8.2 × 120 / 100 = 984 / 100 = 9.84 litres

Therefore,

1 litres = 1.65 dollars

9.84 litres = ?

cross multiply

cost of fuel used = 1.65 × 9.84 / 1

cost of fuel used = $16.236

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