Is the distributive property used to help simplify math expressions

Answer:

16

Step-by-step explanation:

a/(1-r)

Use the infinite geometric series formula.

ANSWER

[tex]S_{\infty} = 16 [/tex]

EXPLANATION

Given the series:

8 + 4 + 2 + . . .

The first term of the series is

[tex]a_1 = 8[/tex]

The common ratio is

[tex]r = \frac{4}{8} = \frac{1}{2} [/tex]

The sum of this series is given by

[tex]S_{\infty} = \frac{a_1}{1 - r} [/tex]

[tex]S_{\infty} = \frac{8}{1 - \frac{1}{2} } [/tex]

Simplify:

[tex]S_{\infty} = \frac{8}{\frac{1}{2} } = 8 \times 2 = 16 [/tex]

Answer:

The Numerator is 6x + 4y - 3z

Step-by-step explanation:

Step  1  :

           z

Simplify   —

           4

Equation at the end of step  1  :

  x    y     z

 (— +  —) -  —

  2    3     4

Step  2  :

           y

Simplify   —

           3

Equation at the end of step  2  :

  x    y     z

 (— +  —) -  —

  2    3     4

Step  3  :

           x

Simplify   —

           2

Equation at the end of step  3  :

  x    y     z

 (— +  —) -  —

  2    3     4

Step  4  :

Calculating the Least Common Multiple :

4.1    Find the Least Common Multiple  

     The left denominator is :       2  

     The right denominator is :       3  

       Number of times each prime factor

       appears in the factorization of:

Prime  

Factor   Left  

Denominator   Right  

Denominator   L.C.M = Max  

{Left,Right}  

2 1 0 1

3 0 1 1

Product of all  

Prime Factors  2 3 6

     Least Common Multiple:  

     6  

Calculating Multipliers :

4.2    Calculate multipliers for the two fractions  

   Denote the Least Common Multiple by  L.C.M  

   Denote the Left Multiplier by  Left_M  

   Denote the Right Multiplier by  Right_M  

   Denote the Left Deniminator by  L_Deno  

   Denote the Right Multiplier by  R_Deno  

  Left_M = L.C.M / L_Deno = 3

  Right_M = L.C.M / R_Deno = 2

Making Equivalent Fractions :

4.3      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent,  y/(y+1)2   and  (y2+y)/(y+1)3  are equivalent as well.  

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

  L. Mult. • L. Num.      x • 3

  ——————————————————  =   —————

        L.C.M               6  

  R. Mult. • R. Num.      y • 2

  ——————————————————  =   —————

        L.C.M               6  

Adding fractions that have a common denominator :

4.4       Adding up the two equivalent fractions  

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

x • 3 + y • 2     3x + 2y

—————————————  =  ———————

      6              6    

Equation at the end of step  4  :

 (3x + 2y)    z

 ————————— -  —

     6        4

Step  5  :

Calculating the Least Common Multiple :

5.1    Find the Least Common Multiple  

     The left denominator is :       6  

     The right denominator is :       4  

       Number of times each prime factor

       appears in the factorization of:

Prime  

Factor   Left  

Denominator   Right  

Denominator   L.C.M = Max  

{Left,Right}  

2 1 2 2

3 1 0 1

Product of all  

Prime Factors  6 4 12

     Least Common Multiple:  

     12  

Calculating Multipliers :

5.2    Calculate multipliers for the two fractions  

   Denote the Least Common Multiple by  L.C.M  

   Denote the Left Multiplier by  Left_M  

   Denote the Right Multiplier by  Right_M  

   Denote the Left Deniminator by  L_Deno  

   Denote the Right Multiplier by  R_Deno  

  Left_M = L.C.M / L_Deno = 2

  Right_M = L.C.M / R_Deno = 3

Making Equivalent Fractions :

5.3      Rewrite the two fractions into equivalent fractions

  L. Mult. • L. Num.      (3x+2y) • 2

  ——————————————————  =   ———————————

        L.C.M                 12      

  R. Mult. • R. Num.      z • 3

  ——————————————————  =   —————

        L.C.M              12  

Adding fractions that have a common denominator :

5.4       Adding up the two equivalent fractions  

(3x+2y) • 2 - (z • 3)     6x + 4y - 3z

—————————————————————  =  ————————————

         12                    12      

Final result :

 6x + 4y - 3z

 ————————————

      12    

For this case we have by definition that:

[tex](f * g) (x) = f (x) * g (x)[/tex]

In this case we must find[tex](t * s) (x),[/tex] so:

([tex](t * s) (x) = t (x) * s (x) = (4x ^ 2-x + 3) * (x-7) =[/tex]

We must apply distributive property that states that:

[tex](a + b) (c + d) = ac + ad + bc + bd[/tex]

So:

[tex](4x ^ 2-x + 3) * (x-7) = 4x ^ 3-28x ^ 2-x ^ 2 + 7x + 3x-21 = 4x ^ 3-29x ^ 2 + 10x-21[/tex]

Answer:

[tex]4x^3-29x^2+10x-21[/tex]

I think it's width= 3 times length

So the 1. length is 18/3=6

The 2. Width is 8*3=24

The 4. width is 12*3=36

The reason is because your mutliplying by 3

Answer:

Interest earned at 3.9 percent rate is $31.2

Interest earned at 2 percent rate is $5.8

Step-by-step explanation:

A = P(1 + rt)

Where 'A' is the amount, 'r' is the rate and 't' is the time in years

When;

P = $1200

r = 3.9%

t = [tex]\frac{8}{12}[/tex] years

Then,

A = $1200(1 + 0.039([tex]\frac{8}{12}[/tex]))

A = $1200 + $31.2 = $1231.2

Interest = Amount - Principal

Interest earned at 3.9 percent rate is $1231.2 - $1200 = $31.2

When;

P = $580

r = 2%

t = [tex]\frac{6}{12}[/tex] years

Then,

A = $580(1 + 0.02([tex]\frac{6}{12}[/tex]))

A = $580 + $5.8 = $585.8

Interest earned = Amount - Principal

Interest earned at 2 percent rate = $585.8 - $580 = $5.8

Thesimple interest, you use the formula I = PRT, with P representing principal, R the annual interest rate in decimal form, and T the time in years. For $1200 at 3.9% over 8 months, the interest is $37.20. For $580 at 2% over 6 months, the interest is $5.80.

The simple interest earned on two separate principals with different interest rates and time periods. To calculate simple interest, we use the formula I = PRT, where I is the interest earned, P is the principal amount, R is the annual interest rate (in decimal form), and T is the time in years. If the time is given in months, we must convert it into years by dividing by 12.

Example 1:
Principal (P) = $1200
Annual Interest Rate (R) = 3.9% = 0.039
Time (T) = 8 months = ({8}{12}) years

Simple Interest (I) = $1200 (times) 0.039 (times) \(frac{8}{12}) = $37.20 to the nearest cent

Example 2:
Principal (P) = $580
Annual Interest Rate (R) = 2% = 0.02
Time (T) = 6 months = ({6}{12}) years

Simple Interest (I) = $580 (times) 0.02 (times) \({6}{12}) = $5.80 to the nearest cent

In conclusion, we have two amounts of simple interest for the different principals, rates, and times provided. To find the total future amount with simple interest, we would add the interest earned to the principal amount.

1st one is 0

2nd one is -3

3rd one is -2

4th one is 5

Combine like terms, isolate the variable, and make the slope of the variable 1 to find all the answers.

Answer:

x + y = 5

Hope this helps you out!

So you work for 2 hours and each hour you get 7.50

So you do 2*7.50 = 15$

Then you do the windows

3 hours and every hour you earn 6.25

3*6.25=18.75$

Now you add them

15+18.75= 33.75$ but if you want an average then around 34$

The first one is three

The second one is twelve

The last one is 2

Answer:

28

Step-by-step explanation:

We start by doing 28 - 14 which equals 14.

We do this because we want to find out the amount of books before he bought 14. We have 14 which is after he sold half of his comics. A half is 1/2, that means you multiply 14 by 2. You get 28.

Final answer:

Julio initially had 28 comic books, sold half, then purchased 14 more, which resulted in him having a total of 28 comic books. We used a simple algebraic equation to determine the original amount.

Explanation:

Let's denote the original number of comic books that Julio had as x. According to the problem, Julio sold half of his comic books (x/2) and then bought 14 more. Consequently, he now has 28 comic books in total.

To express this situation mathematically, we can create the following equation to represent the changes in the number of comic books Julio has:

x/2 + 14 = 28

If we simplify this equation by subtracting 14 from both sides, we get:

x/2 = 28 - 14

x/2 = 14

By multiplying both sides of the equation by 2, we can find the original number of comic books Julio had:

x = 14 * 2

x = 28

Therefore, Julio began with 28 comic books before he sold half and bought 14 more.

Answer:

[tex]x<-2[/tex]

Step-by-step explanation:

Isolate the variable to find the solution set.

[tex]5>x+7 \\ \\ x<-2[/tex]

Answer:

see explanation

Step-by-step explanation:

Solve the inequality

5 > x + 7 ( subtract 7 from both sides )

- 2 > x ⇒ x < - 2

solution set {x | x ∈ R, x < - 2 }

Hello,

First we work out the equations:

x + y =62 will be the first equation.

2x= y +13 is the second equation.

We can first rewrite the second equation as 2x – y =13.

So we have:

x + y = 62

2x –y =13

KEEP IN MIND: With y being positive in one of the equations and negative in the other, we can combine the equations to quickly eliminate y and solve for x.

x + y = 62

+2x –y =13

3x = 75 divide both sides by 3 to get x.

x = 25

Now that we have x we can substitute the value for x, 25.

25 + y = 62 we can subtract 25 from both sides to get y.

y = 62- 25

y = 37

2(25) = 37 + 13

Therefore,

50 = 50

Have a amazing day.

The two numbers your friend has are 25 and 37. This solution is found by creating a system of equations based on the clues given and solving for the two variables.

Your friend essentially gave you a system of two equations to solve where the two numbers are x and y. The first equation from your friend's statement is '2 times the first number is 13 more than the second number', which transforms to 2x = y + 13. The second equation is 'the sum of the two numbers is 62', which converts to x + y = 62.

Solving this system of equations can reveal the two numbers your friend has. We can utilize the substitution method, where we express y from the first equation as y = 2x - 13 and plug it into the second equation. This substitution gives us x + (2x - 13) = 62. This simplifies to 3x - 13 = 62, and by rearranging, we find 3x = 75. Dividing by 3 we get x = 25.

Using the value of x = 25, now we can find y = 2(25) - 13 = 37.

So, your friend's numbers are 25 and 37.

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Answer:

x=1/2, x=-3

Step-by-step explanation:

2x^2+5x-3 = 0

(2x^2+6x)+(-x-3) = 0

2x(x+3)+-1(x+3) = 0

(2x-1)(x+3)=0

2x-1=0, x+3=0

x=1/2, x=-3

feet will be the best way to measure the high jump

Answer:

median, mean , and mode

Step-by-step explanation:

Answer: $33

Step-by-step explanation: Divide 27/9 which is $3

-so you pay $3 for each pack

-then to find how much you need to pay for 11 packs you just multiply 11*3=33

Hello There! The answer is 12w - 12y + 3.

Here's a step by step on how to solve  3(2w-4y+3)+6w-6.

Step 1: Multiply the 3 by the numbers in the parenthesis.

3(2w-4y+3)+6w-6 ⇒ 6w - 12y + 9 + 6w - 6

Step 2: Rearrange the numbers so like terms are next to each other.

6w - 12y + 9 + 6w - 6 ⇒ 6w + 6w - 12y - 6 + 9.

Step 3: Combine like terms. (6w + 6w and - 6 + 9).

6w + 6w - 12y - 6 + 9 ⇒ 12w - 12y + 3

This is your final answer since you cannot simplify any further. I hope this helps and have a great day!

Answer:

12 w - 12 y + 3

Hope this helps :)

Have a great day !

5INGH

Step-by-step explanation:

3 ( 2 w - 4 y + 3 ) + 6 w - 6 =

6 w - 12 y + 9 + 6 w - 6 =

12 w - 12 y + 3

ANSWER

(1,-3)

EXPLANATION

The given equations are:

y = 5x - 8

y = 4x - 7

We equate the two equations to get:

5x - 8=4x - 7

Group similar terms:

5x - 4x=-7+8

Simplify bottom sides

x=1

Put x=-15 into the first equation:

y=5(1)-8

y=-3

(1,-3)

The solution is (1,-3)

Answer:

B

Step-by-step explanation:

Answer:

The correct option is A.

Step-by-step explanation:

The area of a triangle is

[tex]A=\frac{1}{2}\times base\times height[/tex]

From the given figure it is clear that the base of the triangle STR is 6 ft and the height of the triangle 10 ft. So, the area of triangle STR is

[tex]A=\frac{1}{2}\times 6\times 10[/tex]

[tex]A=3\times 10[/tex]

[tex]A=30[/tex]

The area of triangle STR is 30 square ft. Therefore the correct option is A.

Answer:

x=2i/3

All you have to do is just to find the square root.

Answer:

x = ±(2/3)i

Step-by-step explanation:

Subtract 4, divide by 9, and take the square root.

9x^2 = -4

x^2 = -4/9

x = ±√(-4/9) = ±(2/3)i . . . . . . the solutions are imaginary numbers

Answer:

Step-by-step explanation:

Pythagoream theorem applies.

The long wires are [tex]\sqrt{45^2+30^2}m[/tex] long, or about 54m long.

The short ones are [tex]\sqrt{20^2+30^2}m[/tex], or about 36m long. You will need 4 of each, so it's [tex]4* 54m + 4*36m = 540m[/tex] of wire

Answer:

A

Step-by-step explanation:

Given x < - 6

Since less than, then looking for an open circle at - 6 on the number line

If ≤ then closed circle at - 6

Also since less than - 6 we are looking for the arrow pointing to the left indicating values less than - 6 , arrow right indicates greater than - 6

The option is therefore A

Answer:

Part 1) The length of DC is [tex]13\ units[/tex]

Part 2) The measures of the angles in the isosceles triangle are

The base angles are 67.4° and the vertex angle is 45.2°

Step-by-step explanation:

step 1

In the right triangle BDC Find the length of DC

Applying the Pythagoras Theorem

[tex]DC^{2}=BC^{2} +BD^{2}[/tex]

we have

[tex]BC=AB=10/2=5\ units[/tex]

[tex]BD=12\ units[/tex]

substitute

[tex]DC^{2}=5^{2} +12^{2}[/tex]

[tex]DC^{2}=169[/tex]

[tex]DC=13\ units[/tex]

step 2

Find the measures of internal angles in the isosceles triangle ABC

we know that

∠DAC=∠DCA ------> base angles

∠ADC ------> vertex angle

Find the measure of angle DCA

In the right triangle BDC

sin(∠DCA)=BD/DC

substitute the values

sin(∠DCA)=12/13

∠DCA=arcsin(12/13)=67.4°

so

∠DAC=∠DCA=67.4°

Find the measure of angle ∠ADC

Remember that the sum of the internal angles of a triangle must be equal to 180 degrees

so

∠DAC+∠DCA+∠ADC=180°

substitute

67.4°+67.4°+∠ADC=180°

∠ADC=180°-134.8°=45.2°

Answer:

option C

x = 6

y = 4

z = 1

Step-by-step explanation:

Given in the question a matrix in the row echelon form of the augmented matrix.

[tex]\left[\begin{array}{ccc}1&0&+4=-1\\0&1&-1=+3\\0&0&+1=+1\end{array}\right][/tex]

First Equation

x0 + y0 + z(1) = 1

0 + 0 + z = 1

Second Equation

x0 + y(1) + z(-1) = 3

0 + y - z = 3

y -1 = 3

Third Equation

x(1) + y0 + z(4) = -1

x + 0 + 4z = -1

x + 4 = -1

Answer:

Two more people.

Step-by-step explanation:

If 4 people can mow a lawn in 6 hours, then 4 people can mow 1/6 of a lawn in 1 hour.

If 4 people can mow 1/6 of a lawn in 1 hour, then 1 person can mow 1/24 of a lawn in 1 hour.

If 1 person can mow 1/24 of a lawn in 1 hour, then 6 people can mow 1/4 of a lawn in 1 hour.

If 6 people can mow 1/4 of a lawn in 1 hour, then 6 people can mow a lawn in 4 hours.

6 people minus 4 people is 2 people.

There are two more people will be needed to mow the lawn in 4 hours, It is assuming that each person mows at the same rate

The rate of work is defined as the number of units of work completed per unit time.

Let the number of the person are x,

Condition 1;

4 people in 6 hours complete the work, work done in one hour;

1 hour =(1/6) of the total work

Condition 2;

If four individuals can do one-sixth of the total work, one person can do one-quarter of the work in an hour.

1 person do = (1/4) of the total work;

Condition 3;

If one person can do 1/24 of total work

Six persons can do 1/4 of total work in the same amount of time.

If six individuals can mow a quarter of a lawn in an hour, The number of  people will be needed to mow the lawn in 4 hours;

n=6-4

n=2

Hence, there are two more people will be needed to mow the lawn in 4 hours,

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ANSWER

See explanation

EXPLANATION

The point (3,-6) is the fourth quadrant.

In this quadrant only the cosine ratio and the secant ratio are positive.

The remaining four trigonometric ratios are negative.

The diagram is shown in the attachment.

We use the Pythagoras Theorem to find the hypotenuse, h.

[tex] {h}^{2} = {6}^{2} + {3}^{2} [/tex]

[tex] {h}^{2} = 36 + 9[/tex]

[tex] {h}^{2} = 45[/tex]

[tex]h = \sqrt{45} [/tex]

[tex]h = 3 \sqrt{5} [/tex]

[tex] \sin( \theta) = - \frac{opp}{hyp} [/tex]

[tex]\sin( \theta) = - \frac{6}{3 \sqrt{5} } [/tex]

[tex]\sin( \theta) = - \frac{2 \sqrt{5} }{ 5 } [/tex]

[tex]\sin( \theta) = - \frac{2}{ \sqrt{5} } [/tex]

[tex]\ cos(\theta) = \frac{adj}{hyp} [/tex]

[tex]\ cos(\theta) = \frac{3}{3 \sqrt{5} } [/tex]

[tex]\ cos(\theta) = \frac{1}{ \sqrt{5} } [/tex]

[tex]\ cos(\theta) = \frac{ \sqrt{5} }{5} [/tex]

[tex]\tan(\theta)= - \frac{opp}{hyp} [/tex]

[tex]\tan(\theta)= - \frac{6}{3} = - 2[/tex]

[tex]\csc(\theta)= - \frac{hyp}{opp} [/tex]

[tex]\csc(\theta)= - \frac{3 \sqrt{5} }{6} = - \frac{\sqrt{5} }{2} [/tex]

[tex] \sec( \theta) = \frac{hyp}{adj} [/tex]

[tex]\sec( \theta) = \frac{3 \sqrt{5} }{3} = \sqrt{5} [/tex]

[tex]\cot( \theta) = - \frac{adj}{opp} [/tex]

[tex]\cot( \theta) = - \frac{3}{6} = - \frac{1}{2}[/tex]

Answer:

They distort differences between values.

Step-by-step explanation:

Non-zero axes distort differences between values, but they are not always misleading.

It depends on who is looking at them.

Consider the bar graphs below, showing the change in interest rates over time.

A naïve reader might conclude that interest rates had risen sharply (the zero-axis shows that they have remained steady).

However, a professional who specializes in interest rates might find the information in the left-hand graph quite useful.

A professional who specializes in interest rates might find the information in the left-hand graph quite useful.

They distort differences between values.

We have given that, a graph

They distort differences between values,

Non-zero axes distort differences between values, but they are not always misleading.

It depends on who is looking at them.

Consider the bar graphs below, showing the change in interest rates over time.

A naïve reader might conclude that interest rates had risen sharply (the zero-axis shows that they have remained steady).

However, a professional who specializes in interest rates might find the information in the left-hand graph quite useful.

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Answer:

Step 2, he didn't add the values he completed the square with to both sides.

Step-by-step explanation:

When you complete the square, you need to add the values to both sides, not just one.

So when Homer added 49 + 1 or 50 to the left side of the equation, he should've also added 50 to the right side of the equation.

Hence, the equation would now be :

(x-7)^2 + (y+1)^2 = 75

Answer:

(x - 2)(x² + 2x + 4)

Step-by-step explanation:

A difference of cubes factors in general as

• a³ - b³ = (a - b)(a² + ab + b² )

note 8 = 2³ = 8 ⇒ b = 2 , with a = x

Hence

x³ - 8

= x³ - 2³

= (x - 2)(x² + 2x + 2²) = (x - 2)(x² + 2x + 4)

Answer: The rewritten expression would be [tex](x-2)(x^2+2x+4)[/tex]

Step-by-step explanation:

Since we have given that

[tex]x^3-8[/tex]

We need to use the difference of cubes:

As we know the formula of "Difference of cubes":

[tex]a^3-b^3=(a-b)(a^2+b^2+ab)[/tex]

So, it can be written as

[tex](x)^3-(2)^3=(x-2)(x^2+2x+4)[/tex]

Hence, the rewritten expression would be [tex](x-2)(x^2+2x+4)[/tex]

Answer:63Minutes

Step-by-step explanation:

Answer: 63.

Step-by-step explanation: In order to find how fast she runs 6 miles, it'd be convenient to know how fast she runs in 1 mile. So:

8 miles = 84 minutes.

8/8 = 1

84 / 8 = 10.5

6 x 1 = 6

10.5 x 6= 63.