Simplify the expression.

Answer:

10/18

Step-by-step explanation:

U only asked for what is equivalent

Answer:

10/18, 15/27, 20/36, 25/45, 30/54, 35/63, 40/72, 45/81, 50/90, 55/99, 60/108.

Step-by-step explanation:

New equations for a system can be written by interacting (adding or subtracting) the given equations, using substitution to insert an equation into another, or scaling an equation by multiplying it by a constant.

When working with a system of equations, there are multiple ways in which we can write new equations that create equivalent systems. Here are three methods:

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Final answer:

To create equivalent systems of equations, one can use scaling to multiply an equation by a non-zero constant, substitution to replace variables with equivalent expressions, or elimination through addition or subtraction to reduce the number of variables.

Explanation:

When dealing with a system of equations, there are several methods to create equivalent systems, which effectively means rewriting the equations without changing the solution set. One common approach to achieve this is by using the following techniques:

Scaling: Multiplying an entire equation by a non-zero constant. This changes the coefficients but not the solutions.

Substitution: Replacing one variable with an equivalent expression derived from another equation in the system.

Elimination (Addition or Subtraction): Adding or subtracting equations from each other to eliminate one of the variables, effectively reducing the number of variables to solve for.

These methods allow us to simplify complex systems or prepare them for other solving techniques, such as matrix inversion or graphical representation.

The value of a and b in given expression must be [tex]\frac{2}{5} \text { and } \frac{3}{5}[/tex] respectively so that given equality becomes identity.

Need to find the value of a and b in following expression so that following equality will become identity.

[tex]\frac{(x-1)}{(x+1)(x-4)}=\frac{a}{(x+1)}+\frac{b}{(x-4)}[/tex]   ------- eqn 1

Lets Simplify Right hand Side first,

[tex]\frac{a}{(x+1)}+\frac{b}{(x-4)}=\frac{a(x-4)+b(x+1)}{(x+1)(x-4)}[/tex]

[tex]\begin{array}{l}{=\frac{a x-4 a+b x+b}{(x+1)(x-4)}} \\\\ {=\frac{a x+b x-4 a+b}{(x+1)(x-4)}} \\\\ {=\frac{(a+b) x-4 a+b}{(x+1)(x-4)}}\end{array}[/tex]

[tex]=>\frac{a}{(x+1)}+\frac{b}{(x-4)}=\frac{(a+b) x-4 a+b}{(x+1)(x-4)}[/tex]

[tex]\text {On substituting } \frac{a}{(x+1)}+\frac{b}{(x-4)}=\frac{(a+b) x-4 a+b}{(x+1)(x-4)} \text { in equation } 1 \text { we get }[/tex]

[tex]\frac{(x-1)}{(x+1)(x-4)}=\frac{(a+b) x-4 a+b}{(x+1)(x-4)}[/tex]

On multiplying both sides by (x+1)(x-4) we get

[tex]\frac{(x-1)}{(x+1)(x-4)} \times(x+1)(x-4)=\frac{(a+b) x-4 a+b}{(x+1)(x-4)} \times(x+1)(x-4)[/tex]

[tex]\Rightarrow x-1=(a+b) x-(4 a-b)[/tex]

On comparing coefficient of x and constant term separately, we get

a + b = 1 and 4a - b = 1

On adding the two equations we get

a + b + 4a - b = 1 + 1

=> 5a = 2

=> [tex]a = \frac{2}{5}[/tex]

[tex]\text {Substituting } \mathrm{a}=\frac{2}{5} \text { in equation } a+b=1, \text { we get }[/tex]

[tex]\begin{array}{l}{\frac{2}{5}+b=1} \\\\ {\Rightarrow b=1-\frac{2}{5}=\frac{3}{5}}\end{array}[/tex]

So the value of a and b in given expression must be [tex]\frac{2}{5} \text { and } \frac{3}{5}[/tex] so that given equality becomes identity.

Answer:

The amount of numbers that have a G C F of 7 and a product of 490 is 2.

Step-by-step explanation:

On Prime Factorization 490   =   2   x   5   x   7   x   7

G CF = Greatest Common Factor

So, here the possible number who have products as 490 and G C F as 7 is

(a)  2 x 7 = 14  and   5 x 7  = 35

As, 14 and 35 on multiplication,gives 490 as a  product.

And, G C F  of 14 and 3 5 is also 7.

(b)  7  and   5 x 7 x  2  = 70

As, 7 and 70 on multiplication,gives 490 as a  product.

And, G C F  of 7  and 70 is also 7.

⇒ (14, 35) and (7,70) are the only such pairs.

Hence, the amount of numbers that have a G C F of 7 and a product of 490 is 2.

Final answer:

The question seeks the number of number pairs with a GCF of 7 and a product of 490. There are only two pairs that meet the criteria: (7, 70) and (10, 49). Each pair multiplies to 490 and has a GCF of 7.

Explanation:

The question asks for the number of pairs of numbers that have a Greatest Common Factor (GCF) of 7 and a product of 490. First, we must note that any pair of numbers with a GCF of 7 must be multiples of 7. So, we divide 490 by 7 to find another factor that is also a multiple of 7, which gives us 70. Now, we have two pairs of numbers that satisfy the condition: (7, 70) and (10, 49) because each pair when multiplied gives 490, and the GCF of each pair is 7. It's important to note that 490 is equal to 7^2 times 10, which means we won't find more than these two pairs without repeating the same factors in a different order.

Word problem - Jose played soccer for 5 hours every afternoon for 8 days. How many hours did Jose play soccer in all?

So this problem is about Jose and the amount of time he spends playing soccer.

Next, let's pick out what we know and what we want to find out.

We know that Jose played soccer for 5 hours every afternoon for 8 days. We want to find out how many hours of soccer he played in all so we need to multiply.

We multiply the amount of time he spent playing soccer each afternoon which is 5 hours by the number of days which is 8 days.

(8) (5) = 40

This means that Jose played 40 hours of soccer in all.

Answer:

[tex]p (x) = x^{3} - 21x^{2}+ 147x - 343[/tex]

is the required polynomial with degree 3 and p ( 7 ) = 0

Step-by-step explanation:

Given:

p ( 7 ) = 0

To Find:

p ( x ) = ?

Solution:

Given p ( 7 ) = 0 that means substituting 7 in the polynomial function will get the value of the polynomial as 0.

Therefore zero's of the polynomial is seven i.e 7

Degree : Highest raise to power in the polynomial is the degree of the polynomial

We have the identity,

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

Take a = x

        b = 7

Substitute in the identity we get

[tex](x -7)^{3} = x^{3}-3x^{2}(7) +3x(7)^{2} - 7^{3}\\(x -7)^{3} = x^{3}-21x^{2} +147x - 343[/tex]

Which is the required Polynomial function in degree 3 and if we substitute 7 in the polynomial function will get the value of the polynomial function zero.

p ( 7 ) = 7³ - 21×7² + 147×7 - 7³

p ( 7 ) = 0

[tex]p (x) = x^{3} - 21x^{2}+ 147x - 343[/tex]

A polynomial function with degree 3 and P(7)=0 can be written as p(x) = a(x - 7)(x - r2)(x - r3), where r2 and r3 can be any real number and a is any non-zero real number.

The student has asked for a polynomial function with a degree of 3 such that P(7) equals 0. By definition, a polynomial function of degree n with given roots can be written as:

p(x) = a(x - r1)(x - r2)... (x - rn)

Since we have been given that P(7)=0 , we can choose 7 as one root, but the other roots can be chosen arbitrarily. So, we can create a polynomial function like below:

p(x) = a(x - 7)(x - r2)(x - r3)

Where r2 and r3 can be any real number and a is any non-zero real number. For example, if we choose r2=1, r3=2 and a=1. The function would be:

p(x) = 1*(x - 7)(x - 1)(x - 2) = ([tex]x^3 - 10x^2[/tex] + 29x - 14)

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1 + 2 = 21 (reversing numbers)

2 + 3 = 32 (reversing numbers) + 4 = 36

3 + 4 = 43 (reversing numbers)

4 + 5 = 54 (reversing numbers) + 4 =58

Answer:

Step-by-step explanation:

This is a sequence using reversed numbers, and alternating a summing fixed amount.

In the first one you can see that it's only numbers being reversed.

The second one is also reversing numbers but adding 4 units, 32+4=36.

The third one has the same pattern in the first one, just numbers being reversed.

The fourth has the same pattern than the second one, reversed numbers and add 4 units. So, it would be 54+4=58.

Therefore, the answer is 58.

Answer:

3x^3-18x^2+26x-8

Step-by-step explanation:

(x-4)(3x^2-6x+2)

3x^3-6x^2+2x-12x^2+24x-8

3x^3-18x^2+26x-8

Answer:

3x^3-18x^2+26x-8

Step-by-step explanation:

(x-4)(3x^2-6x+2)

3x^3-6x^2+2x-12x^2+24x-8

3x^3-18x^2+26x-8

Answer:

[tex]x \in \mathbb \: R[/tex]

Step-by-step explanation:

The given equation is:

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

We expand to get:

[tex] - 2x - 2 - 1 = - x - x - 3[/tex]

We group like terms to get:

[tex] - 2x + x + x = - 3 + 1 + 2[/tex]

We combine the like terms to obtain:

[tex] - 2x + 2x = - 3 + 3[/tex]

[tex] 0 = 0[/tex]

This is always true therefore the solution is infinite.

Answer: this equation has no solutions ,I just got it correct

Step-by-step explanation:

Answer:

No solutions

Step-by-step explanation:

Khan Academy says its right

Answer:

1,170,000

Step-by-step explanation:

The returns are 15% of 1,300,000.

15% of 1,300,000 = 15% * 1,300,000 = 0.15 * 1,300,000 = 195,000

In 1 month, 195,000 smartphones were returned.

In 6 months, 6 times as many smartphones were returned.

6 * 195,000 = 1,170,000

Answer: 1,170,000

Answer:

1,170,000 Smartphones were returned over a 6-month period.

Step-by-step explanation:

Ok, so to start off, 15% of 1,300,000 is 195,000. Now if everything remains the same over the next 5 months, you just need to multiply the 195,000 by 6 to get the 6 month return rate. This results in the number of 1,170,000 Smartphones Returned over a 6-month period.

The required equation of line is:

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

Step-by-step explanation:

Given equation of line is:

y=-2x-3

The coefficient of x is the slope of line as the equation is in slope intercept form

Let m1 be the slope of given graph of line

So,

[tex]m_1=-2[/tex]

The product of slopes of perpendicular lines is -1

Let m2 be the slope of required line

Then

[tex]m_1.m_2 = -1\\-2 . m_2 = -1\\m_2 = \frac{-1}{-2}\\m_2 = \frac{1}{2}[/tex]

the slope-intercept form of line is:

[tex]y=m_2x+b[/tex]

Putting the value of slope

[tex]y=\frac{1}{2}x+b[/tex]

To find the value of b, putting the given point (-2,-1) in equation

[tex]-1=\frac{1}{2}(-2)+b\\-1 = -1 +b\\-1+1 = b\\b = 0[/tex]

Hence,

The required equation of line is:

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

Keywords: Equation of line, slope-intercept form

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Answer:

10x+15

Step-by-step explanation:

5(2x+3)=10x+15

Answer:

10x+15.

Step-by-step explanation:

Its just simplified.

5 times 2x is 10x

5 times 3 is 15.

Put it together 10x+15.

Answer: OPTION D.

Step-by-step explanation:

The rate of change of a linear function is also known as "Slope".

The slope can be calculated with the following formula:

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

Choose two points on the function "R" graphed:

[tex](1,0)\\(0,-3)[/tex]

You can say that:

 [tex]y_2=-3\\y_1=0\\\\x_2=0\\x_1=1[/tex]

Substituting values into the formula, you get:

[tex]m=\frac{-3-0}{0-1}=3[/tex]

Choose two point from the table that models the function "S":

[tex](4,21)\\(6,31)[/tex]

You can say that:

 [tex]y_2=31\\y_1=21\\\\x_2=6\\x_1=4[/tex]

Substituting values into the formula, you get:

[tex]m=\frac{31-21}{6-4}=5[/tex]

Therefore, you can conclude that the function "S" has a greater rate of change than Function "R".

Answer:

D: Function S has a greater rate of change than Function R.

Step-by-step explanation:

Function S has a greater rate of change than Function R.

Function R → m =

rise

run

=

6

2

= 3. Function S → m =

change in y

change in x

=

31 − 21

6 − 4

=

10

2

= 5

Since 5 > 3, function S has a greater rate of change.

Answer:

x^2-3x-3=0

Step-by-step explanation:

x^2=3x+3

x^2-3x-3=0

Answer:

2 scoops

Step-by-step explanation:

Lester needs to add 2 scoops of flour of 1/3 cup measure.

We have Lester who needs to add 2/3 of a cup of flour but he only has a 1/3 cup measure.

We have to find out how many scoops of flour does Lester need to add.

Assume the number of scoops = x

Then -

10x = 100

x = 10

According to question, we have -

Amount of flour needed by Lester = 2/3

Size of Cup measure = 1/3

Assume that the the number of scoops of protein powder be y.

Then -

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

y = 2

Hence, Lester needs to add 2 scoops of flour 1/3 cup measure.

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Answer:

18

Step-by-step explanation:

I'm 90% sure I'm right but I could be wrong

4 × 18 = 72

6 × 18 = 108

72 + 108 = 180 degrees

Which means the opposite angle diagonal from 6x would also equal 108 degrees

Same thing for 4x, 72 degrees

72° + 72 ° + 108° + 108° = 360 degrees

Sigma notation of sequence −10, −13, −16, … is [tex]\sum_{n=1}^{6}-(3 n+7)[/tex]

Need to determine the sigma notation for the following sequence  

−10, −13, −16, …

Let's try to build a generic formula for given sequence  

The given sequence is in Arithmetic progression where first term = -10 and common difference = -3

The formula for arithmetic progression is given as:

[tex]a_n = a_1 + (n - 1)d[/tex]

Where,

[tex]a_n[/tex] is the nth term in the sequence

[tex]a_1[/tex] is the first term in the sequence

d is the common difference between the terms

Here in this sequence [tex]a_1[/tex] = -10 and d = -3

[tex]a_n = -10 + (n - 1)(-3)\\\\a_n = -10 - 3n + 3\\\\a_n = -3n - 7[/tex]

So generic formula for a term is – ( 3n + 7 )

[tex]\begin{array}{l}{\text { For } \mathrm{n}=1, \text { term is }-(3 \times 1+7)=-10} \\\\ {\text { For } \mathrm{n}=2, \text { term is }-(3 \times 2+7)=-13} \\\\ {\text { For } \mathrm{n}=3, \text { term is }-(3 \times 3+7)=-16}\end{array}[/tex]

And so on

Using sigma notation for arithmetic sequence:

[tex]\sum_{k=1}^{n} a_{k}[/tex]

So for first six terms value of n will vary from 1 to 6 and in sigma notation it can be represented as

[tex]\sum_{n=1}^{6}-(3 n+7)[/tex]

Answer:

D

Step-by-step explanation:

The line representing Bella has a steeper slope than Kofi's, so the slope of that line must be greater than 15.55.

Similarly, the line representing Elsie has a shallower slope than Kofi's, so the slope of that line must be less than 15.55.

The only option that fits is D.

Answer:

Range = {-2, 1, 4, 13}

Step-by-step explanation:

Domain is the set of x-values that are allowed for the function. The values for which the function is defined.

Range is the set of y-values that are allowed for the fucntion. The values for which the function is defined.

Since domain is given as:

Domain = {-1, 0, 1, 4}

We have to plug in each of the 4 values and find the corresponding 4 values for the range. Lets do this below:

y = 3x + 1

y = 3(-1) + 1

y = -2

y = 3x + 1

y = 3(0) + 1

y = 1

y = 3x + 1

y = 3(1) + 1

y = 4

y = 3x + 1

y = 3(4) + 1

y = 13

So, the range would be:

Range = {-2, 1, 4, 13}

In a function, each "x" can only have one "y" number. A relation can still be a function if a y-value has more than one x-value.

Answer:

11. a)

The relationship represented by the graph is a function because it passes the vertical line test (VLT). Any vertical line you pass through the graph, it only "hits" one spot on the graph. For a relation to be a function, each x-value can only have one y-value. This means no hour can have two different amounts of bacteria.

b)

If there were the same number of bacteria for two consecutive hours, the graph would still represent a function. In this case, two values for "y" will have more than one value for "x". Although, each x-value still only has one y-value, so it's still a function.

Answer:

R = 8.31

Step-by-step explanation:

From the table choose a corresponding amount for Volume and pressure.  Plug it into the equation and solve for R.

Choosing the second row V = 16.62 and p = .5

p = R/V

.5=R/16.62

multiply both sides by 16.62 to isolate R

16.62(.5) = R

R = 8.31

Answer:8.31

Step-by-step explanation:

Answer:

1 and 7/8 chapter

Step-by-step explanation:

1 ÷ 1/5 = 5

3/8 x 5 = 15/8 or 1 7/8

Answer:

1 7/8 chapters

Step-by-step explanation:

Answer:

She bought 3 large frames and 14 small frames

The cost of the nylon rope is $0.8 per meter. For $1, you can purchase 1.25 meters of nylon rope.

The given equation is p=0.8L. Here, 'p' stands for the price in dollars and 'L' is the length of the nylon rope in meters.

A. To determine the cost per meter, we look at the coefficient in front of 'L' in our equation. Here, it is 0.8, signifying that the nylon rope costs $0.8 per meter.

B. To find out the length of rope that could be bought for $1, we use the equation by setting 'p' equal to 1 and solving for 'L'. Doing this gives us 'L' = 1 / 0.8 which equals to 1.25 meters.

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Answer:

A. packaging costs

Step-by-step explanation:

Packaging costs woupd change based on how many products are being shipped. The other payments are fixed because they are the same amount paid regularly. Packaging would be considered a variable cost.

Internet access fee is not an example of a fixed cost.

The fixed cost definition states that businesses incur a cost that does not change positively or negatively with the number of goods sold or services given.

A fixed cost is a cost that remains constant regardless of production or sales volume.

Examples of fixed costs include monthly rent, packaging costs, and weekly fixed electricity costs.

Internet access fee is not an example of a fixed cost since it fluctuates based on usage.

The cost of internet access fee will increase or decrease depending on the amount of data used or the duration of usage.

Hence, Internet access fee is not an example of a fixed cost.

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The inequality which shows the number of weeks, w, you need to save to be able to go on a trip is 25w + 200 ≥ 535.

Linear inequalities are defined as those expressions which are connected by inequality signs like >, <, ≤, ≥ and ≠ and the value of the exponent of the variable is 1.

Total amount needed to go for a trip to California = $535

Already saved amount = $200

You decide to save an additional $25 per week.

Let w be the number of weeks needed to save $25 to get the amount needed to go for trip.

These saved amounts must be at least $535 to go for trip.

The amount saved after w weeks is 25w

Total saved amount = 200 + 25w

This saved amount must be ≥ 535.

So the inequality is 200 + 25w ≥ 535.

Hence the inequality which represents the amount need to save to go for trip is  200 + 25w ≥ 535.

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Answer:

B. 10

Step-by-step explanation:

x + y = 21

xy = 110

Rewrite first equation. x + y = 21 becomes y = -x + 21

Now substitute this new equation in the second equation.

x(-x+21) = 110

-x^2 + 21x - 110

Divide by -1

x^2 - 21x + 110

(x - 10)(x - 11)

x = 10, 11

10 is less, so the youngest daughter is 10