Chapter: Simple linear equations (Answer in steps)

Answer:

see explanation

Step-by-step explanation:

The n th term of an AP is

[tex]a_{n}[/tex] = a₁ + (n - 1)d

where a₁ is the first term and d the common difference

Given

6(a₁ + 5d) = 13(a₁ + 12d) ← distribute parenthesis on both sides

6a₁ + 30d = 13a₁ + 156d ( subtract 13a₁ from both sides )

- 7a₁ + 30d = 156d ( subtract 30d from both sides )

- 7a₁ = 126d ( divide both sides by - 7 )

a₁ = - 18d

Now

a₁₉ = a₁ + 18d = - 18d + 18d = 0 ← as required

Answer:

A. 24

Step-by-step explanation:

4/9 = x/54

x= 54*4/9     ===== multiplying both sides by 54

x= 24

Answer is 24, choice A is correct one

Answer: x = {-2, 2}

Step-by-step explanation:

Tangent means it is touching. Find the intersection of the two equations.

Solve the linear equation for y, then set the two equations equal to each other.

[tex]4y=x+8\qquad \rightarrow \qquad y=\dfrac{x+8}{4}[/tex]

[tex]\dfrac{x-1}{x}=\dfrac{x+8}{4}\\\\\\\text{Cross multiply and solve for x:}\\4(x-1)=x(x+8)\\4x-4=x^2+8x\\.\qquad 0=x^2+4x+4\\.\qquad 0=(x+2)^2\\.\qquad 0=x+2\\.\qquad x=-2[/tex]

To find the next point that is parallel to the linear equation and tangent to the curve, we need to use the linear equation with slope (m) = [tex]\dfrac{1}{4}[/tex] and unknown b.

Let's try b = 0, then the equation of the linear equation is: [tex]y=\dfrac{1}{4}x[/tex]

Set the equations equal to each other and solve for x:

[tex]\dfrac{x-1}{x}=\dfrac{x}{4}\\\\\\4(x-1)=x^2\\4x-4=x^2\\.\qquad 0=x^2-4x+4\\.\qquad 0=(x-2)^2\\.\qquad 0=x-2\\.\qquad x=2[/tex]

This works!!!  If it didn't work, we would have tried other values for b until we arrived at a solution.

Step-by-step explanation:

81^x² = 27^x

(3^4)^x² = (3^3)^x

3^(4x²) = 3^(3x)

4x² = 3x

4x² − 3x = 0

x (4x − 3) = 0

x = 0 or ¾

Answer:

converse of alternate exterior angle theorem

Step-by-step explanation:

um im not sure if i should explain the full proof but

Answer:

Below

Step-by-step explanation:

The length of this triangle is 3x+1 and the width is x.

The perimeter P is:

P= 2(3x+1)+2*x

P= 6x+2+2x

P= 8x+2

Let's evaluate it when x=1/2

●1/2 =0.5

P= 8*0.5+2 =4+2= 6 ft

●●●●●●●●●●●●●●●●●●●●●●●●

The area A is:

A = (3x+1)*x

A= 3x^2 +x

Let's evaluate it when x=0.5 feet

A= 3*0.5^2 +0.5

A= 3*0.25+0.5

A= 0.75 +0.5

A= 1.25 ft^2

Answer:

14 panels

Step-by-step explanation:

Area of four walls is given by 2*(length + width)*height

_______________________________________

Given dimension

Length = 16 feet

width = 12 feet

height = 8 feet

Thus, area of four walls of office = 2(16+12)8= 448

_____________________________________________

dimension of paneling sheets

length = 8 feet

width = 4 feet

area of paneling sheets = 8*4  = 32 sq. feet

Let the number of paneling sheets required by n

thus, total area of n paneling sheets = n*area of paneling sheets  = 32n

This, area of paneling sheets (32n) should be same as 448 area of four walls

as given " I have 4 feet by 8 feet paneling sheets for the walls"

thus,

32n = 448

n = 448/32 = 14

Thus, 14 panels are needed.

Answer:

The best first step would be to multiply the second equation by -2

Step-by-step explanation:

The best first step would be to multiply the second equation by -2

then you would have the following

[tex]\ \ \ \ \ \ 4x - 5y = 2 \\-2*(2x)+ (-2)*y = (-2)*3[/tex]

and when you multiply it is easy to eliminate because you will get

   [tex]4x - 5y = 2 \\-4x -2y = 6[/tex]

and if you sum the equations you get

-7y = 8

so that is a single variable equation which is easier to solve.

Answer:

the daily fee =33 dollars

and the mileage charge.=0.35

Step-by-step explanation:

let d: be daily fee and m for mileage

cost of rental =(d*number of days)+ (m*number of mileage)

her first trip: 4d+440m=286

her second trip: 3d+190m=165.5

solve by addition and elimination

4d+440m=286 ⇒ multiply by 3 ⇒12d +1320m=(3)286

3d+190m=165.5⇒ multiply by 4⇒12d+190(4)m=4(165.5)

12d+1320m=858

12d+760m=662

subtract two equation to eliminate d

12d+1320m-12d-760m=858-662

560m=196

m=7/20=0.35 for on mileage

d: 4d+440m=286

4d=286-440(0.35)

d=(286-154)/4 33 dollars

Answer:

It will take Mrs Chandler 18 hours 45 minutes or 18.75 hours to catch up Mr Chandler

Step-by-step explanation:

What we want to know here is that at how many hours will they have traveled same distance.

Let the total time taken by Mrs Chandler to catch up be x hours

Since Mr Chandler left 2.5 hours earlier , then the total time taken by him would be x + 2.5 hours

Now, we know that distance = speed * time

Since it’s same distance covered;

For Mr Chandler, his distance is calculated as 75(x + 2.5)

For Mrs Chandler, her distance is calculated as 85x

We equate both since they are equal;

75(x + 2.5) = 85x

75x + 187.5 = 85x

85x -75x = 187.5

10x = 187.5

x = 18.75 hours or 18 hours 45 minutes

Answer: Perpendicular.

Step-by-step explanation:

Suppose that you have two perpendicular lines:

Remember that a line is something like:

y = a*x +b

and two lines are parallel if they have the same slope (a) but a different y-intercept(b)

Then our lines can be:

y1 = a*x + b1

y2 = a*x + b2.

Now, if we have a line:

y = a*x + b

Then a perpendicular line will have a slope equal to -(1/a):

yp = (-1/a)*x + c

So this only depends on the slope, and we know that our two parallel lines have the same slope. So if we construct a transversal line that is perpendicular to one of our lines, it also must be perpendicular to the other line.

Answer:

A

Step-by-step explanation:

Answer:

-3x^2 -5x +2

Step-by-step explanation:

-2x(x+3)-(x+1)(x-2)=

Distribute

-2x^2 -6x  -(x+1)(x-2)

Foil

-2x^2 -6x  -(x^2 -2x +x -2)

Combine like terms

-2x^2 -6x  -(x^2 -x  -2)

Distribute the minus sign

-2x^2 -6x  -x^2  +x +2

Combine like terms

-2x^2  -x^2  -6x +x +2

-3x^2 -5x +2

Answer:

[tex]\huge\boxed{-2x(x+3)-(x+1)(x-2)=-3x^2-5x+2}[/tex]

Step-by-step explanation:

[tex]-2x(x+3)-(x+1)(x-2)[/tex]

Use the distributive property: a(b + c) = ab + ac

and FOIL: (a + b)(c + d) = ac + ad + bc + bd

[tex]=(-2x)(x)+(-2x)(3)-\bigg[(x)(x)+(x)(-2)+(1)(x)+(1)(-2)\bigg]\\\\=-2x^2-6x-\bigg(x^2-2x+x-2\bigg)=-2x^2-6x-x^2-(-2x)-x-(-2)\\\\=-2x^2-6x-x^2+2x-x+2[/tex]

Combine like terms:

[tex]=(-2x^2-x^2)+(-6x+2x-x)+2=-3x^2+(-5x)+2\\\\=-3x^2-5x+2[/tex]

Step-by-step explanation:

The center of a circle is the point in the circle which is equidistant to all the edges of thr circle. The point a is the center, while point b is an arbitrary point in the circle. Find attachment for the diagram.

Answer:

i think that this question is wrong

Step-by-step explanation:

Answer/Step-by-step Explanation:

To create a stem-and-leaf plot for the GPAs of the 20 students that were listed in the above question, take the following steps:

Step 1: for easy plotting, write down the GPAs in an ordered manner, that is, from the smallest value to the largest.

0.5, 0.5, 0.9, 1.1, 1.6, 1.6, 1.7, 1.8, 1.9, 1.9, 2.1, 2.3, 2.4, 2.9, 3.3, 3.4, 3.6, 4.0,  4.0,  4.0

Step 2: divide each set of data into a stem and a leaf. For example, for a GPA, 0.5, 0 would be the stem, while 5 would be the leaf.

For a GPA listed, 2.9, 2 is the stem, 9 is the leaf. Same applies to all the other GPAs.

Step 3: All stems should be written in ascending order, vertically, from the smallest to the largest. From the listed GPAs, you would observe that the highest stem value would be 4, while the lowest would be 0.

Therefore, write down your stems vertically, in ascending order as shown below:

0 |

1 |

2 |

3 |

4 |

Step 4: All leaves should be written also in ascending order to their corresponding stem, from the smallest to the largest, as shown below:

Stem | Leaf

0 | 5 5 9

1 | 1 6 6 7 8 9 9

2 | 1 3 4 9

3 | 3 4 6

4 | 0 0 0

Thus,

0 | 5 5 9 represents => 0.5, 0.5, 0.9

*See attachment below for the stem-and-leaf plot.

Answer:

(-2,-3)

Step-by-step explanation:

Well in the system,

x−5y=13

4x−3y=1

We need to find x or y in either equation.

Let's do x - 5y = 13 for x.

+5y to both sides

x = 5y + 13

Now we substitute 5y + 13 for y in 4x - 3y = 1.

4(5y + 13) - 3y = 1

20y + 52 - 3y = 1

17y + 52 = 1

-52 to both sides

17y = -51

Divide all by 17

y = -3

Now we can substitute -3 for y in 4x - 3y = 1.

4x - 3(-3) = 1

4x + 9 = 1

-9 to both sides

4x = -8

Divide 4 to both sides

x = -2

Thus,

the solution is (-2,-3).

Hope this helps :)

Answer:

Step-by-step explanation:

x - 5y = 13

4x - 3y = 1

Solve the equation for x

[tex]x - 5y = 13[/tex]

Move '5y' to R.H.S and change it's sign

[tex]x = 13 + 5y[/tex]

Substitute the given value of X into the equation

4x - 3y = 1

[tex]4(13 + 5y) - 3y = 1[/tex]

Solve the equation for y

distribute 4 through the parentheses

[tex]52 + 20y - 3y = 1[/tex]

Collect like terms

[tex]52 + 17y = 1[/tex]

Move constant to R.H.S and change it's sign

[tex]17y = 1 - 52[/tex]

Calculate

[tex]17y = - 51[/tex]

Divide both sides of the equation by 17

[tex] \frac{17y}{17} = \frac{ - 51}{17} [/tex]

Calculate

[tex]y = - 3[/tex]

Now, substitute the given value of y into the equation

x = 13 + 5y

[tex]x = 13 + 5 \times ( - 3)[/tex]

Solve the equation for x

Multiply the numbers

[tex] = 13 - 15[/tex]

Calculate the difference

[tex] = - 2[/tex]

The possible solution of the system is the ordered pair

( x , y )

( x , y ) = ( - 2 , - 3 )

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

Check if the given ordered pair is the solution of the system of equation

[tex] - 2 - 5 \times ( - 3) = 15[/tex]

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

Simplify the equalities

[tex]13 = 13[/tex]

[tex]1 = 1[/tex]

Since all of the equalities are true, the ordered pair is the solution of the system

Hope this helps..

Best regards!!

Answer:

Option E

Step-by-step explanation:

y = x /3

let x = 1, 2, 3

y = 0.333, 0.667, 1

y = x/2 + 40

let x = 1, 2, 3

y = 40.5, 41, 41.5

y = x

let x = 1, 2, 3

y = 1, 2, 3

y = 2x + 50

let x = 1, 2, 3

y = 52, 54, 56

y = 3x - 20

let x = 1, 2, 3

y = -17, -14, -11

The standard deviation is the spread of data, the data that is most spread is option E.

Answer:

The slope is -4.

Step-by-step explanation:

The values -2 and -6 are 4 values apart.

The values -4 and -3 are 1 value apart.

Since the second coordinate is lower than the first one, the slope of this is negative.

4 / 1 = 1

Negating 1 gets us -1.

Hope this helped!

Answer:

[tex] \frac{y}{x} = \frac{ - 4}{1} = - 4[/tex]

Step-by-step explanation:

[tex]x = ( - 3) - ( - 4) = 1[/tex]

[tex]y = ( - 6) - ( - 2) = - 4[/tex]

Answer:

To verify the Cauchy-Bunyakovsky-Schwarz Inequality, (p,q) must be less than (or equal to) ||p|| • ||q||

(1,1,1) is not equal to (-10,5)

Step-by-step explanation:

a°b° + a^1b^1 + a^2b^2 < 5x (-2x^2 + 1)

Any algebra raised to the power of zero is equal to 1.

a°b° = 1 × 1 = 1

1 + ab + a^2b^2 < -10x^3 + 5x

The vectors:

(1,1,1) < (-10,5)

This verifies the Cauchy-Schwarz Inequality

Triangle Inequality states that for any triangle, the sum of the lengths of two sides must be greater than or equal to the length of the third side.

Answer:

a)  [tex]\frac{2}{3} \,\frac{lb}{bread}[/tex]

b)  [tex]1\frac{1}{4} \,\frac{in}{domino}[/tex]

Step-by-step explanation:

Part a:

every 4 lbs of flour, she makes 6 loaves of bread. this as a rate in simplest fraction form is:

[tex]\frac{4}{6} \,\frac{lb}{bread} = \frac{2}{3} \,\frac{lb}{bread}[/tex]

Part b:

every 10 inches , 8 dominoes can be placed. then the rate can be written as:

[tex]\frac{10}{8} \,\frac{in}{domino} = \frac{5}{4} \,\frac{in}{domino} =1\frac{1}{4} \,\frac{in}{domino}[/tex]

Answer:

Hello  your questions is incomplete attached below is the missing part of the question

answer : A )  0.647 ,  (B) 0.353,  (C) students are more likely to spend the money than to have kept it

Step-by-step explanation:

from the attached table below

Given data :

Total number of students = Number of students who spent money + number of students who kept money

Total number of students = (33+13 ) + (18 + 27 ) =  91

p(Number of students given four quarters) = (33 +18 ) / 91 = 51/91

p( number of students who spent money ) = ( 33 +13 ) / 91 = 46/91

p( number of students who saved money ) = (18 +27 ) / 91 = 45 /91

p( number of students who spent money and given four quarters ) = 33/91

p( number of students who saved money and given four quarters ) = 18/91

A) The probability of randomly selecting a student who spent the money and also given four quarters

= p ( 33/91 | 51/91 )

= 33/91 * 91/51

= 33/51 = 0.647

B ) The probability of randomly selecting a student who kept the money and given that the student was given four quarters

= p ( 18/91 | 51/91 )

= 18/91 * 91/51

= 18 /51 = 0.353

C) students are more likely to spend the money than to have kept it

Answer:

For 3^-6 = 1/3^6 = 726 and for 3^-4 = 1/3^4 = 81

Step-by-step explanation:

ARE THE NUMBERS SEPARATED OR THEY ARE TO BE JOIN TOGETHER,IF THERE ARE SEPARATED THE SOLUTION IS GOING TO BE...

3^-6

using law of indices x^-a = 1/x^a

So 3^-6 = 1/3^6

Now find the power of 3^6

which is the same as 3×3×3×3×3×3 = 729

therefore 1/3^6 = 729

For 3^-4

using the above method

3^-4 is same as 1/3^4

which is equal to

finding the nominator

3×3×3×3 = 81

so the answer is 81

Answer:

y = -4x or the second option on edge.

This is because after you form it into the given equation, it equals y = -4x.

In order to clarify, edge also states that's the answer.

Answer:

2nd option

Step-by-step explanation:

Answer:

there were 63,000L of water initially in the pool

Step-by-step explanation:

do 900L times the first 20 minutes to find out how much water was drained during the first 20 minutes and you get 18,000L drained

then, add 18,000L plus the rest of the 45,000L of water drained from the pool to get 63,000L of water intially in the pool

Answer:

x = 5

Step-by-step explanation:

x = 8 - 3

Thus, x = 5

Answer:

y = 6x - 26

Step-by-step explanation:

1. find slope: (y₂ - y₁) / (x₂ - x₁)

(-2 - 10) / (4 - 6) = -12 / -2 = 6

basic equation: y = 6x + b

2. plug in (x,y) value using one set of coordinates.

10 = 6(6) + b

10 = 36 + b

b = 10 - 36

b = -26

3. plug b in to find full equation.

y = 6x -26

Answer:

y = -1/6 x + 11

Step-by-step explanation:

In order to write an equation of a line you need slope (m) and y-intercept (b) or where the graph grosses the y- axis. since you are given two points (6, 10) and (4, -2). Slope when given two points is (y - y) / (x - x)

                                                                  so (-2 - 10) / (6 - 4) = 6 / - 1 =- 6

use the equation y = mx + b and substitute either point (6, 10) or (4, -2) as a replacement for x and y respectively. (I chose (6, 10) because they are positive numbers. Substituting x = 6 and y = 10 and m = -6  into y = mx + b

10 = -6(6) + b

10 = -36 + b

b = 46 (add -36 to both sides)

so our equation: y = -6x + 46 :-)

Answer:

B. The company should have 8 production runs each year.

Step-by-step explanation:

From the given information:

A paint manufacturer has a uniform annual demand for 16,000 cans of automobile primer

It costs $4 to store one can of paint for one year

$500 to set up the plan for production of the primer

Let x be the number of cans of paint produced during each production run

Let y be the number of production runs.

If the total storage and setup cost is C = 500y + 2c

and cy = 16000

Then c = 16000/y

From;

C = 500y + 2c

Replacing c with 16000/y,  we have;

C = 500y + 2(16000/y)

C = 500y + 32000/y

in order to minimize the total storage and setup costs  [tex]C_{min} = C[/tex]

Therefore [tex]\dfrac{dc}{dy}=0[/tex]

⇒ 500 - 32000/y² =0

y² = 32000/500

y² = 320/5

y² = 64

y = (√64)

y = 8

Therefore; The company should have 8 production runs each year in order to minimize the total storage and setup costs

Answer:

6.3 ft

Step-by-step explanation:

The ladder lying on the wall with an elevation of 39° forms a right angled triangle.

Hypotenuse = length of ladder = 10 ft

Opposite = x = ?

θ = 39°

Use trigonometric ratio formula to find, x, which is how far the ladder goes up the wall.

The trigonometric ratio formula to use is:

sin(θ) = opposite/hypotenuse

Sin(39) = x/10

Multiply both sides by 10

10*sin(39) = x

x = 10*sin(39)

x = 6.29 ≈ 6.3 ft (to nearest tenth)

Answer:

6.3 ft

Step-by-step explanation:

did the quiz got it right

Answer:

Step-by-step explanation:

The statement

The value of y varies inversely as the square of x is written as

[tex]y = \frac{k}{ {x}^{2} } [/tex]

where k is the constant of proportionality

To find the value of x when y = 1 first find the formula for the variation

y = 16 x = 3

k = yx²

k = 16(3)²

k = 16 × 9

k = 144

The formula for the variation is

[tex]y = \frac{144}{ {x}^{2} } [/tex]

when y = 1

We have

[tex]1 = \frac{144}{ {x}^{2} } [/tex]

Cross multiply

x² = 144

Find the square root of both sides

We have the final answer as

Hope this helps you

Answer:

Step-by-step explanation:

Substitute n = 1, n = 2, n = 3 and n = 4 to the equation f(n) = n² - 1:

f(1) = 1² - 1 = 1 - 1 = 0

f(2) = 2² - 1 = 4 - 1 = 3

f(3) = 3² - 1 = 9 - 1 = 8

f(4) = 4² - 1 = 16 - 1 = 15

Answer:

f(x)= -3x + 400

Step-by-step explanation:

[tex]\frac{x-x_{1} }{x_{2}-x_{1} } = \frac{y-y_{1} }{y_{2}-y_{1} }[/tex]

[tex]\frac{x-3}{13-3} =\frac{y-391}{361-391}[/tex]

-3 ( x-3 ) = (y - 391 )

-3x + 400

Answer:

he is correct

Step-by-step explanation: