use the elimination method to solve the system of equations. 6x+4y=32 -6x+4y=8

Answer:

[tex]\large\boxed{1.\ \dfrac{1}{x}=5+2\sqrt6}\\\boxed{2.\ \dfrac{x-1}{x}=-4-2\sqrt6}\\\boxed{3.\ \dfrac{x+1}{x}=6+2\sqrt6}[/tex]

Step-by-step explanation:

[tex]x=5-2\sqrt6\\\\1.\\\\\dfrac{1}{x}=\dfrac{1}{5-2\sqrt6}=\dfrac{1}{5-2\sqrt6}\cdot\dfrac{5+2\sqrt6}{5+2\sqrt6}\qquad\text{use}\ (a-b)(a+b)=a^2-b^2\\\\=\dfrac{5+2\sqrt6}{5^2-(2\sqrt6)^2}=\dfrac{5+2\sqrt6}{25-2^2(\sqrt6)^2}=\dfrac{5+2\sqrt6}{25-(4)(6)}=\dfrac{5+2\sqrt6}{25-24}\\\\=\dfrac{5+2\sqrt6}{1}=5+2\sqrt6\\\\2.\\\\\dfrac{x-1}{x}=\dfrac{x}{x}-\dfrac{1}{x}=1-\dfrac{1}{x}\\\\\text{use the value of}\ \dfrac{1}{x}\ \text{from 1.}\\\\\dfrac{x-1}{x}=1-(5+2\sqrt6)=1-5-2\sqrt6=-4-2\sqrt6[/tex]

[tex]3.\\\\\dfrac{x+1}{x}=\dfrac{x}{x}+\dfrac{1}{x}=1+\dfrac{1}{x}\\\\\text{use the value of}\ \dfrac{1}{x}\ \text{from 1.}\\\\\dfrac{x+1}{x}=1+5+2\sqrt6=6+2\sqrt6[/tex]

Answer:

25

Step-by-step explanation:

Here we are supposed to find the distance between two coordinates ( 4,10) and (-3,-14).

Let us do this step by step with detailed explanation.

We will use distance formula in order to find the distance between them. The distance formula is given as

[tex]D=\sqrt{(x_{2}-x_{1})^{2} +(y_{2}-y_{1})^{2} }[/tex]

Here we have

[tex]x_{2}=-3\\x_{1}=4\\y_{2}=-14\\y_{1}=10\\[/tex]

Replacing them in the formula we get

[tex]D=\sqrt{(-3-4)^{2} +(-14-10)^{2} }\\D=\sqrt{(-7)^{2} +(-24)^{2} }\\D=\sqrt{49+ 576 }\\D=\sqrt{625 }\\D=25\\[/tex]

Hence our distance is 25 units

Final answer:

The distance between the points (4,10) and (-3,-14) on the coordinate plane is 25 units, calculated using the distance formula.

Explanation:

The distance between the points (4,10) and (-3,-14) on the coordinate plane can be found using the distance formula, which is derived from the Pythagorean theorem. The distance formula is √((x2 - x1)² + (y2 - y1)²), where (x1, y1) and (x2, y2) are the coordinates of the two points.

To calculate the distance:

The distance between the points is 25 units.

ANSWER

[tex]y = - 2{x}^{2} -28x - 96[/tex]

EXPLANATION

We have that

[tex]x = - 8 \: \: and \: \: x = - 6[/tex]

are the roots of the quadratic function.

This implies that

[tex]x + 8 \: \: and \: \: x + 6[/tex]

are factors of the quadratic function.

The quadratic function will have an equation of the form:

[tex]y = a(x + 8)(x + 6)[/tex]

It was also given that, the vertex of the function is at

[tex](-7, 2)[/tex]

This point must satisfy the equation.

This implies that:

[tex]2= a( - 7 + 8)( - 7+ 6)[/tex]

This implies that,

[tex]2=-a[/tex]

[tex]a = - 2[/tex]

We substitute the value of 'a' to get the equation in factored form as:

[tex]y = - 2(x + 8)(x + 6)[/tex]

We expand the parenthesis to write the equation in standard form.

[tex]y = - 2( {x}^{2} + 6x + 8x + 48)[/tex]

[tex]y = - 2( {x}^{2} + 14x + 48)[/tex]

[tex]y = - 2{x}^{2} -28x - 96[/tex]

Or in vertex form, the equation is

[tex]y = - 2 {(x + 7)}^{2} + 2[/tex]

[tex]x^2-x-90=0\\x^2+9x-10x-90=0\\x(x+9)-10(x+9)=0\\(x-10)(x+9)=0\\x=10 \vee x=-9[/tex]

It is formed by two rays lie in a plane.

Answer:

3√3

Step-by-step explanation:

r = 3 cos θ

r = 2 - cos θ

First, find the intersections.

3 cos θ = 2 - cos θ

4 cos θ = 2

cos θ = 1/2

θ = -π/3, π/3

We want the area inside the first curve and outside the second curve.  So R = 3 cos θ and r = 2 - cos θ, such that R > r.

Now that we have the limits, we can integrate.

A = ∫ ½ (R² - r²) dθ

A = ∫ ½ ((3 cos θ)² - (2 - cos θ)²) dθ

A = ∫ ½ (9 cos² θ - (4 - 4 cos θ + cos² θ)) dθ

A = ∫ ½ (9 cos² θ - 4 + 4 cos θ - cos² θ) dθ

A = ∫ ½ (8 cos² θ + 4 cos θ - 4) dθ

A = ∫ (4 cos² θ + 2 cos θ - 2) dθ

Using power reduction formula:

A = ∫ (2 + 2 cos(2θ) + 2 cos θ - 2) dθ

A = ∫ (2 cos(2θ) + 2 cos θ) dθ

Integrating:

A = (sin (2θ) + 2 sin θ) |-π/3 to π/3

A = (sin (2π/3) + 2 sin(π/3)) - (sin (-2π/3) + 2 sin(-π/3))

A = (½√3 + √3) - (-½√3 - √3)

A = 1.5√3 - (-1.5√3)

A = 3√3

The area inside of r = 3 cos θ and outside of r = 2 - cos θ is 3√3.

The graph of the curves is:

desmos.com/calculator/541zniwefe

The area of the region inside r=3cos(\theta) and outside r=2-cos(\theta) is obtained by integrating the square of each function times 1/2 over their intersection interval and subtracting the results.

The student is tasked with finding the area of a region bounded by two polar curves r=3cos(\theta) and r=2-cos(\theta). This involves sketching the curves to identify the area that lies inside the first curve and outside the second one. To find the area of the region, we calculate the difference between the integrals of the two functions over the interval where they intersect. This requires setting up and evaluating definite integrals in polar coordinates. The integral calculation would involve integrating the function r^2/2 from the lower to the upper bound of \(\theta\) for each curve and then subtracting the area inside r=2-cos(\theta) from the area inside r=3cos(\theta).

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Answer:

d

Step-by-step explanation:

the angle is less than 90 degrees

Answer:

the correct choice is an acute angle have a great day

Step-by-step explanation:

To find the slope and y-intercept of the equation -y = -0.9x + 4, multiply both sides by -1 to get y = 0.9x - 4, revealing a slope of 0.9 and a y-intercept of -4.

To find the slope and y-intercept of the graph -y = -0.9x + 4, we first need to rewrite the equation in the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.

By multiplying both sides of the equation by -1, we get y = 0.9x - 4. Here, it's clear that the slope (m) is 0.9 and the y-intercept (b) is -4.

This means that for every increase of 1 on the horizontal axis (x), there is a rise of 0.9 on the vertical axis (y). Additionally, the point where the line intersects the y-axis is at y = -4.

Answer:

x = 12

Step-by-step explanation:

Assuming you reqire the value of x that makes

4 : x equivalent to 5 : 15

Divide 5 by 4, that is 5 ÷ 4 = 1.25

Thus 1.25x = 15 ( divide both sides by 1.25 )

x = 12

Hence

4 : 12 = 5 : 15

Answer:

[tex](2y-1)^{2}[/tex]

Step-by-step explanation:

[tex]4y^{2} -4y+1[/tex]

Sum = -4

Product = 1

Factors = -1 , -1

[tex](2y-1)^{2}[/tex]

Answer:

$13.95

Step-by-step explanation:

You need to multiply the 5 notebooks with the costs of each notebook, which gives you the total costs before tax.

a quadratic equation is an equation with a degree of 2.

well, is not 2x + x + 3 for sure, and is not 3x³ + 2x + 2 either, that one is a cubic, 3rd degree.

well, one would think is 0x² - 4x + 7, however 0x² is really 0, anything times 0 is 0, so the deceptive equation is really -4x + 7, which is not a quadratic.

5x² - 4x + 5 on the other hand is.

Answer:

x = 2.6667 (rounded)

Step-by-step explanation:

Move the constant to the right side and change its sign.

3x = 7 + 1.

Add the numbers.

3x = 8.

Divide both sides by 3.

x = 8/3 = 2.6667 (Rounded to the nearest ten-thousandth).

Hope this helps.

The solution of the equation, rounding the answer to the nearest ten-thousandth, is x = 2.6667.

To find the solution of the equation 3x - 1 = 7, we can follow these steps:

Step 1: Add 1 to both sides of the equation to isolate the term with x.

3x - 1 + 1 = 7 + 1

3x = 8

Step 2: Divide both sides of the equation by 3 to solve for x.

3x/3 = 8/3

x = 8/3

Step 3: Convert the fraction 8/3 to a decimal.

x ≈ 2.6667

Rounding the decimal to the nearest ten-thousandth, the solution of the equation is approximately x = 2.6667.

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Answer:

-2/7

Step-by-step explanation:

1. simplify the right side of equation by mult

2. divide by seven

3. whatever is being multiplied by x is the slope

please MARK ME BRAINLIEST

The slope of the linear equation 7Y= -2(X - 4) will be negative 2/7.

A relationship between two or more parameters that, when shown on a graph, produces a linear model. The degree of the variable will be one.

The linear equation is given as,

y = mx + c

Where m is the slope of the line and c is the y-intercept of the line.

The linear equation is given below.

7Y= -2(X - 4)

Simplify the equation in the slope-intercept form will be

7Y= -2X + 8

Y = (-2/7)X + 8/7

Then the slope of the linear equation 7Y= -2(X - 4) will be negative 2/7.

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Answer:

Ay=3 over 2 or y=-2 over 3

Step-by-step explanation:

the quadratic formula: y={-b±√(b²-4ac)}/2a

In the equation 6y²-5y-6= 0, a=6, b=-5, c= -6

Substituting for the values in the formula we get:

{-(-5)±√[(-5²)-4(6)(-6)}/2(6)

{5±√169}12

={5±13}/12

(5+13)/12=3/2 or (5-13)/12= -2/3

Answer:

[tex]y=\frac{3}{2}[/tex] or [tex]y=-\frac{2}{3}[/tex]

Step-by-step explanation:

We have the expression

[tex]6y^2-5y-6 = 0[/tex]

For an equation of the form [tex]ay^2 +by +c[/tex]  the quadratic formula is

[tex]y=\frac{-b \± \sqrt{b^2 -4ac}}{2a}[/tex]

In this case

[tex]a = 6\\b= -5\\c =-6[/tex]

[tex]y=\frac{-(-5) \± \sqrt{(-5)^2 -4(6)(-6)}}{2(6)}[/tex]

[tex]y_1=\frac{3}{2}[/tex]

[tex]y_2=-\frac{2}{3}[/tex]

Answer:

Step-by-step explanation:

A graph shows that only the points listed above are within the doubly-shaded area. Points on the boundary line are not solutions, since the inequalities do not include the "or equal to" case.

Answer:

x = - 5

Step-by-step explanation:

Using the rule of exponents

• [tex]a^{-m}[/tex] ⇔ [tex]\frac{1}{a^{m} }[/tex]

Given

[tex](1/2)^{x}[/tex] = 32

[tex]\frac{1}{2^{x} }[/tex] = 32

[tex]2^{-x}[/tex] = [tex]2^{5}[/tex]

Since the bases are equal, both 5, equate the exponents

Hence x = - 5

Answer:

f(n)=8(-3)^(n-1)

The formula which can be used to show the nth term of a geometric sequence where the first term is 8 and the common ratio is - 3 is,

⇒ - 8/3 × (- 3)ⁿ

An sequence has the ratio of every two successive terms is a constant, is called a Geometric sequence.

We have to given that;

The first term is, 8

And, the common ratio is –3

Now,

The nth term of a geometric sequence is:

⇒ T (n) = arⁿ⁻¹

Put a = 8 and r = - 3

⇒ T (n) = 8 (- 3)ⁿ⁻¹

⇒ T (n) = - 8/3 × (- 3)ⁿ

Thus, The formula which can be used to show the nth term of a geometric sequence where the first term is 8 and the common ratio is - 3 is,

⇒ - 8/3 × (- 3)ⁿ

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Answer: 4x^4

Step-by-step explanation:

Firstly, let's apply the ^1/4 to the 256

The fourth root of 256 is 4, that will be the coefficient in our answer

Next, let's apply the ^1/4 to the x^16

The power of powers property says we can multiply the two exponents as so:

(x^16)^1/4 = x^(16*1/4) = x^4

Now combine the 4 and x^4 to get: 4x^4

Answer:

1/150 is greater than 1/200. A general rule is that the larger the denominator is, the smaller the fraction.

The bigger fraction is,

⇒ 1 / 150

A fraction is a part of whole number, and a way to split up a number into equal parts. Or, A number which is expressed as a quotient is called fraction. It can be written as the form of p : q, which is equivalent to p / q.

We have to given that;

Two numbers are,

⇒ 1/150

⇒ 1/200

Since, We know that;

⇒ 1 / 150 = 0.0067

⇒ 1 / 200 = 0.005

Hence, The bigger fraction is,

⇒ 1 / 150

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Answer:

Part A

to the 10th power will be positive and to the 11th power is negative. When the exponent is even it will be positive. When it is odd it will be negative

Part B

The one with the negative enclosed will be positive. The second one, the exponent is only effecting the number and not the sign. The outcome will always be negative.

Answer:

2568 kg

Step-by-step explanation:

30816 divided by 24 gives us 1284 or the total number of cartons which can then be multiplied by 2 to give us the weight of the empty cartons.

To find the total mass of the empty cartons, divide the total number of exercise books by the number of exercise books in each carton and then multiply by the difference in mass between a full and empty carton.

To find the total mass of the empty cartons, we need to determine how many cartons there are and the difference in mass between a full carton and an empty carton.

Given that each carton contains 24 exercise books and there are 30816 exercise books in total, we can calculate the number of cartons by dividing the total number of exercise books by 24.

So, there are 1284 cartons in total.

Now, we need to find the difference in mass between a full carton and an empty carton. The difference is 12 kg - 2 kg = 10 kg.

To find the total mass of the empty cartons, we multiply the number of cartons by the mass difference: 1284 cartons * 10 kg/carton = 12840 kg.

Therefore, the total mass of the empty cartons is 12840 kg.

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The student's mathematics problem requires solving a quadratic equation derived from the statement that the square of '9 less than a number' equals '3 less than the number'. By expanding, rearranging, and factoring the quadratic equation, we find that the number in question is 12.

The student's question involves solving an algebraic equation to find an unknown number.

Let's denote the unknown number as x.

The problem states that the square of 9 less than the number is 3 less than the number itself.

This can be represented as the equation:

(x - 9)^2 = x - 3

To solve for x, we will follow these steps:

Therefore, the two possibilities for the unknown number are 7 and 12.

We must check which one satisfies the original equation, and we find that x = 12 is the correct solution.

=

Answer:

7/6 is the answer, or as a mixed fraction, 1 1/6

Step-by-step explanation:

Turn 2 1/3 into an improper fraction. This gives 7/3. Since dividing by 3 is the same as multiplying by 1/2, 7/3 * 1/2 multiply top and bottom with each other to get 7/6 as your answer.

Hope this helps!

Answer:

[tex]\frac{7}{9}[/tex]

Step-by-step explanation:

2[tex]\frac{1}{3}[/tex] ÷3

= [tex]\frac{7}{3}[/tex] ÷3

= [tex]\frac{7}{3}[/tex] x [tex]\frac{1}{3}[/tex]

= [tex]\frac{7}{9}[/tex]

Answer:

43091.3 or 43181.81(81 repeated)

Step-by-step explanation:

There are two answers because it depends on the method you use. You can get the second answer by plugging 95 into the equation gram*0.0022=pound or gram=(pound/0.0022). When you insert the 95, you get 43181.81(81 repeated).You can find the first answer by just looking it up on the conversion chart on google.

Answer:

43091.275 Grams

Step-by-step explanation:

95 pounds is equal to 43091.275 grams.

Multiply the mass value by 453.592

let's say the point dividing JK is say point P, so the JK segment gets split into two pieces, JP and PK

[tex]\bf ~~~~~~~~~~~~\textit{internal division of a line segment} \\\\\\ J(-25,10)\qquad K(5,-20)\qquad \qquad \stackrel{\textit{ratio from J to K}}{7:3} \\\\\\ \cfrac{J~~\begin{matrix} P \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}{~~\begin{matrix} P \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~K} = \cfrac{7}{3}\implies \cfrac{J}{K} = \cfrac{7}{3}\implies3J=7K\implies 3(-25,10)=7(5,-20)\\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf P=\left(\frac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \frac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)\\\\[-0.35em] ~\dotfill\\\\ P=\left(\cfrac{(3\cdot -25)+(7\cdot 5)}{7+3}\quad ,\quad \stackrel{\textit{y-coordinate}}{\cfrac{(3\cdot 10)+(7\cdot -20)}{7+3}}\right) \\\\\\ P=\left( \qquad ,\quad \cfrac{30-140}{10} \right)\implies P=\left(\qquad ,~~\cfrac{-110}{10} \right)\implies P=(\qquad ,\quad -11)[/tex]

Answer:

-11

Step-by-step explanation:

J =(–25, 10)

K=(5, –20)

Point L divides JK into a 7:3 ratio

To find the coordinates of L we will use section formula.

Formula : [tex]x=\frac{mx_2+nx_1}{m+n}[/tex] and [tex]y=\frac{my_2+ny_1}{m+n}[/tex]

m: n = 7: 3

[tex](x_1,y_1)=(-25,10)\\(x_2,y_2)=(5,-20)[/tex]

Substitute the values

[tex]x=\frac{7(5)+3(-25)}{7+3}[/tex] and [tex]y=\frac{7(-20)+3(10)}{7+3}[/tex]

[tex]x=-4[/tex] and [tex]y=-11[/tex]

Hence the y coordinate of L is -11

To simplify the expression 3−9 · 36 · 36, we understand that 36 is 3 raised to the power of 6. Multiplying 36 by itself means adding the exponents, resulting in 3¹². Subtracting 9 (the minus sign before 9) from 12 (the exponent in 36 · 36), we get 3³, which is equal to 27.

To simplify the expression 3−9 · 36 · 36, we first need to understand the rules for cubing of exponentials and the operations of exponents.

To cube an exponential, you cube the base digit and multiply the exponent by 3.

For example, when cubing 3², we raise 3 to the power of 6 because the original exponent is multiplied by 3 (2×3=6), which gives us 3⁶, or in other words, 729.

In this case, however, we need to simplify the original expression.

We'll start by recognizing that 36 is 3 raised to the power of 6.

The expression states to multiply 36 by itself, which is essentially stating 3⁶ × 3⁶.

When multiplying exponents with the same base, we simply add the exponents, in this case, 6 + 6, which gives us 3¹².

Hence, the expression can be simplified to 3−9 · 3¹².

Now we are subtracting exponents (because of the minus sign), which means we take 9 away from 12 giving us 3³, or 3 cubed, which equals 27.

Answer:

(2x)^2-(5)^2

Step-by-step explanation:

Answer:

(3x + 5)(4x - 7)

Step-by-step explanation:

The most common factors of 35 are 5 and 7. The 12 is a nuisance.

We could try 6 and 2 or 3 and 4 or even 12 and 1. You just have to juggle a bit to get the difference to be - x.  

I'm going to try 3 and 4 first with 5 and 7. The numbers you get for the middle term have to be quite close.

(3x     5)(4x      7) Oops. It's going to come out first try.

Now you need to get the middle term come to - 1

(3x   -  5)(4x  +    7)

Just to make it slightly harder, I'll do it the wrong way first.

-5*4x + 3x*7 = - 20x + 21x = x That is close. Only the signs are incorrect.

So try again.

(3x + 5)(4x - 7)

20x - 21x = - x