Solve the system of equations: 3 � - � = 17 5 � + 3 � = 5

We can factor [tex]$(x^2 y^2-z^2)^2-4x^2y^2$[/tex]as:

[tex]$(x^2 y^2-z^2)^2-4x^2y^2 = (x^2 y^2-z^2-2xy\sqrt{2})(x^2 y^2-z^2+2xy\sqrt{2})$[/tex]

To calculate this:

We can see that each factor is of degree 1 and has a positive coefficient of x.

In each factor, the variables appear in alphabetic order, with the constant term at the end.

In order to factor this expression, we first use the difference of squares, by breaking it down [tex]$(x^2 y^2-z^2)^2$[/tex] as [tex]$(x^2 y^2-z^2)(x^2 y^2-z^2)$[/tex]

We can then add and subtract a suitable term, here we added and subtract 2xy * √2, to make one of the terms in each of the brackets match one of the terms of the original expression, so we can factor it out.

So, [tex]$(x^2 y^2-z^2)^2-4x^2y^2 = (x^2 y^2-z^2-2xy\sqrt{2})(x^2 y^2-z^2+2xy\sqrt{2})$[/tex].

Factors are the numbers that divide into a given number without leaving a remainder.

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The expression for the side of the square will be s=x^(1/2) in

where s=side.

what are squares?

A square is a two-dimensional plane figure with four equal sides and all four angles are equal to 90 degrees. The properties of the rectangle are somewhat similar to a square, but the difference between the two is, a rectangle has only its opposite sides equal. Therefore, a rectangle is called a square only if all its four sides are of equal length.

The most important properties of a square are listed below:

Area of square=(side)²

Perimeter of square=4*side

Now,

Area of square=(side)²

and given that

A=x in^2

s^2=x

s=√x

Therefore, expression in rational exponent form will be

s=x^(1/2)

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The point (-17, -87) belongs to the third quadrant

From the question, we have the following parameters that can be used in our computation:

Point = (-17, -87)

In the above point, we have

x = -17 and y = -87

This means that the x and the y coordinates are negative

The quadrant where the x and the y coordinates are negative is the third quadrant

Hence, the point is in the third quadrant

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Answer:

...

Step-by-step explanation:

To find the number of calories Davis burns while running for 30 minutes, we can use the information that he burns 1080 calories running for 2.5 hours.

We know that 30 minutes is 1/120 of 2.5 hours. So, we can divide the total number of calories by 120 and we get the calorie burn for 30 minutes:

1080 calories / 120 = 9 calories

The number of calories Davis burns while swimming for 30 minutes can be determined by using the equation y = 266x, where x represents the number of hours he swims. We know that he swims for 30 minutes, or 1/120 hours, so we can plug that value into the equation:

y = 266(1/120) = 2.216 calories

So, Davis burns 7 calories more running for 30 minutes than swimming for 30 minutes, assuming the rates remain constant.

a) The maximum height of a firework is 292.25 feet.

b) The time required by a firework to reach its maximum height is 2.875 second.

Herein we find the case of a firework being launched under free fall motion, that is, a vertical uniformly accelerated motion due to gravity, represented by following quadratic equation:

f(t) = - 16 · t² + 92 · t + 160

Where:

Part A - The maximum height can be found by modifying the quadratic equation into vertex form. This can be done by completing the square:

f(t) = - 16 · t² + 92 · t + 160

f(t) = - 16 · [t² - (23 / 4) · t - 10]

f(t) - 16 · (529 / 64) = - 16 · [t² - (23 / 4) · t + 529 / 64] + 160

f(t) - 292.25 = 16 · (t - 2.875)²

The firework reaches a maximum height of 292.25 feet.

Part B - The firework takes a time of 2.875 seconds to reach its maximum height.

Free fall equation has several mistakes and was corrected according to principles of free fall motion.

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Answer:

10.56

Step-by-step explanation:

Answer:

y= - 2,0

x=0,-5

slope would be y=MX+Y

so -2= -5(0)+0

The remainder of the division is (x-3)(x+13)/x-7

From the information given, we have that;

This is represented as;

2x² + 7x - 39/x - 7

Now, factorize the numerator

Find the pair factors of -78 that sum up to 7, then substitute the value, we have;

2x² + 13x - 6x - 39/x- 7

(2x² + 13x) - (6x-39)/x-7

Factor the common terms

x(x + 13) - 3(x + 13)/x-7

Then, we get;

(x-3)(x+13)/x-7

Hence, the value is (x-3)(x+13)/x-7

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Answer:

y=-4x-24

Step-by-step explanation:

draw the line on a grid paper so that you know the slope must be negative. then plug in the numbers.

Answer:

38

Step-by-step explanation:

True, Deciding that a process that fills bottles with soda is functioning properly by checking the weights for a sample of bottles is an example of inferential statistics.

A subset of statistics known as inferential statistics uses a variety of analytical techniques to extrapolate conclusions about population data from sample data.

It is an example of inferential statistics because you are using a sample of data to make inferences about the population of bottles being filled, rather than measuring the weight of every bottle.

Hence, Deciding that a process that fills bottles with soda is functioning properly by checking the weights for a sample of bottles is an example of inferential statistics.

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A real - life representation of 12 / 49 = 0. 24 is the cost of a single donut in a 49 pack of donuts that cost $ 12.00

The equation is 12 / 49 = 0. 24. A real - life representation of this could have to do with unit cost, and total cost.

The 12 could be the total cost of a package of donuts that have 49 donuts inside.

The unit cost of each of these donuts would therefore be:

= Total cost of donuts / Number of donuts in package

= 12 / 49

= $ 0.24 per donut

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When the prism is taken out of the cylinder, the composite shape's volume is 3.048.

The volume of an item in three dimensions is the area that is enclosed by its boundaries. The space that an object occupies in three dimensions is measured by its volume, which is given in cubic units. A few of illustrations of cubic units are cm3 and in3. On the other side, an item's mass is used to measure how much matter it contains. One typical method to ascertain an object's mass is to weigh it, either in pounds or kilogrammes. Unlike the primary equation for the square of a rectangular box, which would be length, breadth, and height, the primary equation in capacity is length, width, and height.

given

volume of cylinder - 9.234

volume of the prism - 6.186

the volume of composite shape is = volume of cylinder - volume of prism

= 9.234 - 6.186

= 3.048

When the prism is taken out of the cylinder, the composite shape's volume is 3.048.

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49 is the shaded part of the rectangle.

Therefore, the rule for its area is base1+base2/2xh

Base1=9

Base2=20

Height = 4

Therefore, 9+20/2 x 4= 49.

The denominator is the total number of pieces that make up the whole, which is eight. That is the figure beneath the fraction bar. The numerator is the number of shaded portions, which is 7. That is the value shown above the fraction bar.

The shaded region’s area is the difference between the area of the complete polygon and the area of the polygon’s unshaded portion. In polygons, the area of the shaded component can arise in two ways. The darkened zone might be found in the middle of a polygon or on its sides.

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The area of the shaded region of the rectangle is 130 sq. meter

A quadrilateral with four right angles is a rectangle. It may alternatively be described as a parallelogram with a right angle or an equiangular quadrilateral, where equiangular denotes that all of its angles are equal. A square is a rectangle with four equally long sides.

The denominator is the total number of pieces that make up the whole, which is eight. That is the figure beneath the fraction bar. The numerator is the number of shaded portions, which is 7. That is the value shown above the fraction bar.

The shaded region’s area is the difference between the area of the complete polygon and the area of the polygon’s unshaded portion. In polygons, the area of the shaded component can arise in two ways. The darkened zone might be found in the middle of a polygon or on its sides.

The base length = 9 m

The base width = 20 m

The height = 4m

The area of the shaded region = 1/2 *(9 + 4)*20 = 130 sq. meter

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The formula for g(x) is 2√(x-6) + 1. The solution has been obtained using the concept of translation.

What is translation?

In mathematics, a translation involves moving a shape up, down, left, or right. The translated shapes appear to be exactly the same size as the original ones; thus, the shapes are consistent with one another. Just one or more directions have been changed. There is no change to the shape because it is simply being moved from one location to another.

The object's shift in location can occur in a variety of ways, such as initially moving left, then turning right, and so on. Each point on the form will translate by the same number of units.

We are given f(x) = 2√x

Now, shifted down 1 unit, we get

2√x + 1

Also, shifted right 6 units, we get

2√(x-6)

So, g(x) = 2√(x-6) + 1

Hence, the formula for g(x) is 2√(x-6) + 1.

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The measure of angle POQ is given as follows:

30º.

The area of a circle of radius r is given by π multiplied by the radius squared, as follows:

A = πr²

The radius of the circle in this problem is given as follows:

r = OP = 2.

(distance of the center O to point P on the circumference).

Hence the area of the circle is given as follows:

A = 4π.

The area of the sector is given as follows:

π/3.

The ratio of the area of the sector and the area of the circumference is given as follows:

(π/3)/(4π) = 1/12.

The entire area has a measure of 360º, hence the angle measure is obtained as follows:

1/12 x 360 = 30º.

The problem is given by the image presented at the end of the answer.

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Answer:

Step-by-step explanation:

7x9=63

because 7 down times 9 across

a) A procedure that could be run to test how many times out of 100 a fair die should land on the number 1 is given as follows: Select a random number out of six 100 times, with 1 constituting a success and the remaining five numbers constituting a failure.

b) The 95% confidence interval for the true proportion of rolls that will land on the number 1 is given as follows: (0.174, 0.346). The interpretation is that we are 95% sure that the proportion of all rolls that land in one is between these two values.

c) As the lower bound of the interval is above the expected proportion of 0.1667, the confidence interval supports Vy's suspicions that the die is not fair.

A probability is calculated as the division of the number of desired outcomes by the number of total outcomes.

A die has six sides, one out of which is one, hence the probability of a one being rolled is given as follows:

p  = 1/6 = 0.1667.

Hence 100 trials of a procedure with 6 results should be used, in which:

A confidence interval of proportions has the bounds given by the rule presented as follows:

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which the variables used to calculated these bounds are listed as follows:

The confidence level is of 95%, hence the critical value z is the value of Z that has a p-value of [tex]\frac{1+0.95}{2} = 0.975[/tex], so the critical value is z = 1.96.

The parameters for this problem are given as follows:

[tex]n = 100, \pi = \frac{26}{100} = 0.26[/tex]

Then the lower bound of the interval is calculated as follows:

[tex]0.26 - 1.96\sqrt{\frac{0.26(0.74)}{100}} = 0.174[/tex]

The upper bound is calculated as follows:

[tex]0.26 + 1.96\sqrt{\frac{0.26(0.74)}{100}} = 0.346[/tex]

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The volume of the solid with the given semi-circular base with radius 5 units is equal to 333.33 cubic units.

As given in the question,

Radius of the semicircular base = 5 units

Equation of the circle is given by :

x² + y² = r²

⇒ x² + y² = 5²

⇒ x² + y² = 25

⇒ x = √25 - y²

Cross section is perpendicular to the base and it is parallel to the diameter  are squares:

Diameter is double of the radius

s = 2x

 = 2√25 - y²

Volume of the given solid is equal to :

V = [tex]\int\limits^5_0 {s^{2} } \, dy[/tex]

  = [tex]\int\limits^5_0 {( 2\sqrt{25 - y^{2} }) ^{2} } \, dy[/tex]

 = [tex]\int\limits^5_0 {( 4({25 - y^{2} }) } \, dy[/tex]

= 100y - 4y³/3 (for limit 0 to 5)

=  ( 500 - 500/3 ) - 0

= 500 ( 1 - 1/3 )

= 500( 2/3 )

= 1,000/3

= 333.33 cubic units.

Therefore, the volume of the given solid with the given measures is equal to 333.33.

The above question is incomplete, the complete question is :

Use the general slicing method to find the volume of the following solid.

The solid with a semicircular base of radius 5 whose cross sections perpendicular to the base and parallel to the diameter are squares. Place the semicircle on the xy-plane so that its diameter is on the x-axis and it is centered on the y-axis. Set up the integral that gives the volume of the solid. Use increasing limits of integration.

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The following expressions have length dimensions: a. at b. x3/t

d. √2ax

a. at: has a dimension of m because it is the product of acceleration (m/s2) and time(s). b. 1/2at2: has a dimension of m because it is the product of acceleration (m/s2) and time(s) squared. c. x3/t: has a dimension of m because it is the product of distance x (m) cubed and time(s).

e. v2/a: is a product of velocity (m/s) squared and acceleration (m/s2) thus it doesn't have a dimension of length. d. 2ax: is a product of 2, distance x (m), and acceleration (m/s2) so it has dimension of m.

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Yes, cropping a picture and then using a scale factor to resize it is a two-step process.

The image is first cropped by subtracting 2 inches from the right side, altering the image's proportions.

The image is then enlarged using a scale factor of 1.5, which increases the image's dimensions by 1.5.

The final measurements of the image would be its original measurements less 2 inches (for cropping), then multiplied by 1.5. (for the resizing).

It's vital to keep in mind that both picture cropping and image scaling change the size of the final product.

Yes, cropping a photo, then resizing it with a scale factor, is a two-step process.

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Hey there!

Answer:

An equivalent to the given expression [tex]5f+4f[/tex] would be [tex]9f[/tex].

Hope this helps!

Answer:I hope this helps

Step-by-step explanation:

5f+4f

Collect like terms by adding their coefficients
which is like this

(5f+4f)f and now we

Add the numbers in the parentheses

=9f

By Assuming the denominator does not equal zero. a³b³/a⁴b equivalent expression is  b⁴/a.

What is an equivalent expression?

Equivalent expressions are expressions that have the same result when they are simplified.

Two expressions are considered equivalent if they can be transformed into each other by applying mathematical operations such as simplification, combining like terms, or applying the properties of arithmetic and algebra.

Equivalent expressions can be used to solve problems more efficiently or to help understand complex mathematical concepts.

For example, simplifying an expression to its simplest form can make it easier to understand or perform further calculations with it.

Equivalent expressions can also be used to check the correctness of a solution to a problem, by verifying that the answer can be transformed into the expected result.

The given expression can be simplified as follows:

a³b³ / a⁴b = (a³/a⁴) * (b³/b)

a³/a⁴ = 1/a

b³/b = b²

Hence, By Assuming the denominator does not equal zero. a³b³/a⁴b equivalent expression is  b⁴/a.

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The true statement is that the sample mean is not biased

From the question, we have the following parameters that can be used in our computation:

Sample selection of 2

The sample is selected such that all 4 samples are equally likely

This means that

P(Each selection) = 1/4

An unbiased sample is such that reflects all members of the population and in this case, this sample reflects the population

Hence, the sample is unbiased

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The average rate of change on the interval [1, b] is:

r = 4*(b + 1)

For a function f(x), we define the average rate of change on the interval [a, b] as:

r = ( f(b) - f(a))/(b - a)

Here we have the function:

f(x) = 4x² - 7

And the interval is [1, b]

Then the average rate of change is:

r = (f(b) - f(1))/(b - 1)

r = ( 4b² - 7 - 4 + 7)/(b - 1)

r = (4b² - 4)/(b - 1)

r = 4*(b² - 1)/(b - 1)

r = 4*(b + 1)*(b - 1)/(b - 1)

r = 4*(b + 1)

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Therefore , the solution of the given problem of gradient comes out to be

It has been established that line BC and AB are parallel option 2 is correct.

Pick two points on a graph that has a straight line as the direction. Change in y-coordinate divided by change in x-coordinate is the gradient of the line.In a visual representation, parallel lines encircle one another like train tracks. Parallel lines AB and CD go side by side.

Here,

Two lines that are parallel are denoted by the symbol ||. To indicate that AB is parallel to CD, we write AB||CD.

As a result, line segment AC's length is equal to the sum of line segments AB and BC.

The product of AB's and BC's slopes must be negative because AB and BC are obviously perpendicular to one another.

It has been established that line BC and AB are parallel.

Therefore , the solution of the given problem of gradient comes out to be

It has been established that line BC and AB are parallel.

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Since Dean ran 2.3 fewer kilometers than Sam. If Dean ran 6.8 km, the true statements about the distance which Sam ran are:

A. You know the difference in the distances the boys ran, so this is a subtraction problem.

E. The correct equation is s – 2.3 = 6.8.

In Mathematics, distance can be defined as the total amount of ground that is covered or travelled by a physical object (body) over a particular period of time and speed, irrespective of its direction, starting point or ending point.

Since Dean ran 6.8 km and 2.3 fewer kilometers than Sam, we can logically deduce that this is a subtraction problem. Assuming the variable s represent the distance which Sam ran, Sam's distance can be calculated as follows;

s – 2.3 = 6.8

s = 6.8 + 2.3

s = 9.1 kilometers.

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Answer:

[tex]a. \;\; \boxed{\$ 530.60}[/tex]

[tex]b.\; \;\boxed{ \$552.45}[/tex]

[tex]c,\;\;\boxed{\$610.60}[/tex]

[tex]d. \;\; \boxed{\$745.91}[/tex]

Step-by-step explanation:

All four cases deal with compound interest on the same amount of $500 and same interest rate of 2%.

The only difference is in the frequency of compounding and time

Compound Interest Formula

[tex]\boxed{A = P\left(1 + \dfrac{r}{n}\right)^{n\cdot t}\\\\}[/tex]

where

In this particular problem we have
P = $500
r = 2% = 0.02

These are common for all parts of the question
Only n an t are different for each of the question sub-parts

Part a

Compounding is done annually (once a year) for 3 years

n = 1
t = 3 years
n · t = 3

[tex]A = 500\left(1 + \dfrac{0.02}{1}\right)^3\\\\A = 500\left(1.02\right)^3\\\\A = 500(1.061208)\\\\A=\boxed{\$ 530.60}[/tex]

For accuracy of calculations, I will not compute and store the exponent part, I will perform the calculations in one shot

Part b

Here the compounding is done quarterly (4 times a year) for 5 years

n = 4
t = 5 years
nt = 4 · 5 = 20

[tex]A = 500\left(1 + \dfrac{0.02}{4}\right)^{20}\\\\\\A = 500(1.005)^{20}\\\\A =\boxed{ \$552.45}[/tex]

Part c

Compounding done monthly(12 times a year) for 10 years

n = 12

t = 10

nt = 120

[tex]A = 500\left(1 + \dfrac{0.02}{12}\right)^{120}\\\\\\A = \boxed{\$610.60}[/tex]

Part d

First let's figure out what continuous compounding means

[tex]\fbox{\begin{minipage}[t]{1\columnwidth \fboxsep - 2\fboxrule}%\textsf{What is continuous compounding?} \\\textsf{Continuous compounding is the mathematicallimit that compound interest can reach if it's calculated and reinvestedinto an account's balance over a theoretically infinite number ofperiods. While this is not possible in practice, the concept of continuouslycompounded interest is important in finance. (Investopedia)}\}%\end{minipage}}[/tex]

The formula for continuous compounding can be determine by using the standard formula for periodic compounding and taking limits as
[tex]n \rightarrow \infty[/tex]

Therefore, for compounding continuously , the formula can be derived from
[tex]\lim _{n\to \infty } P\left(1 + \dfrac{r}{n}\right)^{nt}\\\\[/tex]

One of the limit formulas states

[tex]\lim _{x\to \infty } \left(1 + \dfrac{a}{x}\right)^{x} = e^a\\\\[/tex]

Therefore
[tex]\lim _{n\to \infty } \left(1 + \dfrac{r}{n}\right)^{n} = e^r\\\\[/tex]

So for the continuous compounding case, the formula is
[tex]\boxed{A = P \cdot e^{rt}}[/tex]

Here we have
r = 0.02
t = 20 years
rt = 20(0.02) = 0.4

Plugging in P = 500, and rt = 0.4 we get
[tex]A = 500 \cdot e^{0.4}\\\\A = \boxed{\$745.91}[/tex]

Answer:

a) $530.60

b) $552.45

c) $610.60

d) $745.91

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Compound Interest Formula}\\\\$ A=P\left(1+\frac{r}{n}\right)^{nt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $n =$ number of times interest is applied per year \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}[/tex]

Given:

Substitute the given values into the compound interest formula and solve for A:

[tex]\implies A=500\left(1+\dfrac{0.02}{1}\right)^{1 \cdot 3}[/tex]

[tex]\implies A=500\left(1.02\right)^{3}[/tex]

[tex]\implies A=500(1.061208)[/tex]

[tex]\implies A=530.604[/tex]

Therefore, the value of the account after 3 years if interest is compounded annually is $530.60 (nearest cent).

Given:

Substitute the given values into the compound interest formula and solve for A:

[tex]\implies A=500\left(1+\dfrac{0.02}{4}\right)^{4 \cdot 5}[/tex]

[tex]\implies A=500\left(1.005\right)^{20}[/tex]

[tex]\implies A=500(1.10489557...)[/tex]

[tex]\implies A=552.447788...[/tex]

Therefore, the value of the account after 5 years if interest is compounded quarterly is $552.45 (nearest cent).

Given:

Substitute the given values into the compound interest formula and solve for A:

[tex]\implies A=500\left(1+\dfrac{0.02}{12}\right)^{12 \cdot 10}[/tex]

[tex]\implies A=500\left(1.0016666...\right)^{120}[/tex]

[tex]\implies A=500(1.22119943...)[/tex]

[tex]\implies A=610.599716...[/tex]

Therefore, the value of the account after 10 years if interest is compounded monthly is $610.60 (nearest cent).

[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Continuous Compounding Interest Formula}\\\\$ A=Pe^{rt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\\phantom{ww}$\bullet$ $P =$ principal amount \\\phantom{ww}$\bullet$ $e =$ Euler's number (constant) \\\phantom{ww}$\bullet$ $r =$ annual interest rate (in decimal form) \\\phantom{ww}$\bullet$ $t =$ time (in years) \\\end{minipage}}[/tex]

Given:

Substitute the given values into the continuous compounding interest formula and solve for A:

[tex]\implies A=500e^{0.02 \cdot 20}[/tex]

[tex]\implies A=500e^{0.4}[/tex]

[tex]\implies A=500(1.49182469...)[/tex]

[tex]\implies A=745.912348...[/tex]

Therefore, the value of the account after 20 years if interest is compounded continuously is $745.91 (nearest cent).

If a northbound train and a southbound train meet each other on parallel tracks heading in opposite directions. The speeds that the  two trains are traveling is:61 mph.

Speed of westbound train = x mph

Speed of eastbound train = x+16

2.5x + 2.5(x+16) = 265

2.5x + 2.5x + 40 = 265

Combine like terms

5x = 225

Divide both side by 5x

x =225/5

x = 45 mph

Westbound train = 45mph

Eastbound train =45 mph +16 mph = 61 mph

Therefore the speed is 61mph.

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