Consider the equation . x-3=2 1/2
(a) List the three related equations
(b) Choose the related equation that isolates the variable and simplify.

By using two equations, 'Thomas's age = Huilan's age + 15' and 'Thomas's age + Huilan's age = 33', and substituting Huilan's age, we find that Thomas is 24 years old.

To solve this problem, we use a system of linear equations. Based on the given information, we can create two equations:

Replacing Huilan's age in the second equation with '(Thomas's age - 15)', we get: Thomas's age + (Thomas's age - 15) = 33

This simplifies to 2 * Thomas's age = 48, meaning Thomas's age = 24 years.

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Answer:

Table B shows a proportional relationship.

Step-by-step explanation:

In a proportional relationship two quantities vary directly with each other. It means

[tex]y\propto x[/tex]

[tex]y=kx[/tex]

Where, k is the constant of variation.

The ordered pairs of table A are (-2,2), (-1,3), (0,0) and (1,5).

From these ordered pairs we can conclude that the value of y-coordinate is not changing according to the x-coordinate because the values of x increased by 1 for each ordered pair but the value of y is not increasing in the same proportion..

The ordered pairs of table B are (-1,-3), (0,0), (1,3) and (2,6). The value of y increasing at a constant rate 3 and the value of y-coordinate is 3 times of x-coordinate.

Choose any two ordered pairs of table B. Let the two points are (0,0) and (1,3), then the constant of variation is

[tex]k=\frac{y_2-y_1}{x_2-x_1}=\frac{3-0}{1-0}=3[/tex]

The proportional relationship is defined as

[tex]y=3x[/tex]

Therefore, 3 is the constant of variation and rate of change.

So, Table B shows a proportional relationship.

Answer:

Table B Shows proportional relationship.

Step-by-step explanation:

Correct answers are the green and yellow dots. Hope this helps. The red dot I got wrong.

Step-by-step explanation:

1. See the attachment for the filled-in diagram. Adding the contents of the figure gives the sum at the bottom, matching selection C.

2. If we let "d" represent the length of the second volyage, then the total length of the two voyages is ...

... (d+43) + d = 1003

... 2d = 960 . . . . . . . subtract 43

... d = 480 . . . . . . . . divide by 2

The second voyage lasted 480 days.

3. 1.9% - 1.9/100 = 0.019. Adding this fraction to the original means the original is multiplied by 1 +0.019 = 1.019. Doing this multiplication each year for t years means the multiplier is (1.019)^t.

Since the starting value (in 1975) is 4 (billion), the population t years after that is ...

... P(t) = 4(1.019)^t

Multiplying it out using the distributive property, you have ...

... x^4 -(x^4 -7x^2 +7x^2 -49)

... = x^4 -x^4 +7x^2 -7x^2 +49 . . . . distribute the minus sign

... = x^4(1 -1) +x^2(7 -7) +49 . . . . . . collect like terms

... = 0 +0 + 49 . . . . . . . . . . . . . . . . . .simplify

... = 49

50 people

Let x represent the number of people on the cruise. The amount they each must pay is ...

... ($58 -(x -42)) = $100 -x

The revenue from the group is the product of the number of people and the amount each pays:

... r(x) = x·(100 -x)

This describes a downward-opeing parabola with zeros at x=0 and x=100. The vertex (maximum) will be found halfway between those zeros, at x=50.

A group size of 50 maximizes revenue from the cruise.

Answer:

He can have 6 friends eat cake

Step-by-step explanation:

If each friend can eat 1/6 of the cake, we need to find how many friend can have a piece

1/6 * f = 1 cake

Multiply each side by 6

6*1/6 * f = 1 *6

f = 6

He can have 6 friends eat cake

Answer:

6 friends can have a piece of cake.

Step-by-step explanation:

Each friend eats 1/6 of the cake.

There are six 1/6's in a unit.

6 friends can have a piece of cake.

Answer:

XY = 150

Step-by-step explanation:

AB = 8

W Z = 80

SF = 10

8 x 10 = 80

So if BC = 15

XY = 15 x 10 = 150

Answer:

Step-by-step explanation:

you can solve with a proportion

8 : 15 = 80 : x

x = 15 * 80 : 8

x = 150

or you find the rate from AD and WZ

8 : 80

simplify

1 : 10

15 * 10 = 150

Answer:

27 ^6

Step-by-step explanation:

The first step is to split the expression into 2 parts  (ab)^x = a^x b^x

(3a^2)^3

3^3  a^2 ^3

27 a^2^3

We can then us the power of power rule to simplify the exponents x^a^b = x^(ab)

27 a^(2*3)

27 ^6

To find the solution to the system of linear equations, set the equations equal to each other and solve for x. Then substitute this value back into either of the equations to find y.

To find the solution to the system of linear equations, we can set the equations equal to each other:

58x + 18 = -12x - 1

Combining like terms, we get:

70x = -19

Divide both sides by 70:

x = -19/70

Substitute this value back into either of the equations to find y:

y = 58(-19/70) + 18 = -10.2

Therefore, the solution to the system of linear equations is x = -19/70 and y = -10.2.

f(x) = 3·2^x

When x=0, any exponential term will have a value of 1, so the y-intercept is the multiplier of the exponential function. Here, it is 3.

When x=1, the exponential term will have a value equal to its base, so the multiplier just found will be multiplied by the base value. Here, f(1) = 3·2, so the base of the exponential term is 2.

Given these considerations, the function is ...

... f(x) = 3·2^x

_____

Comment on notation

The caret (^) is used to signify an exponent. When the exponent consists of anything other than a single number or variable, it must be put in parentheses: 2^(1/2), for example.

The expressions you have written all look like linear functions.

Answer:

c)No, P is the distance from A to B, and Q is the distance from B to A.

Step-by-step explanation:

Point P partitions the directed segment from A to B into a 1:3 ratio.

Ratio is 1:3

So AP is 1  and PB is 3

Q partitions the directed segment from B to A into a 1:3 ratio.

Ratio is 1:3

So BQ is 1  and QA is 3.

That is AQ= 3  and BQ= 1

The ratio of Q and P varies

AP =1  and AQ=3

So P  and Q  are not at the same point.

Because P is the distance from A  to B  and Q is the distance from B to A

Answer:

c

Step-by-step explanation:

bc

Answer:

41 words per minute

Step-by-step explanation:

You just need to divide words by minutes and that will give you words per minute.

410 words / 10 minutes = 410 / 10 words per minute = 41 words per minute

3x = -4y

To solve for X, divide both sides by 3:

x = -4y / 3

The answer is D.

b(3) = -16

b(2) = b(1) -7 = -2 -7 = -9

b(3) = b(2) -7 = -9 -7 = -16

Answer: b(3)= -16

Step-by-step explanation:

Answer:

x=2

Step-by-step explanation:

3(3x+10)=50−x

The first step is to distribute the 3

3*3x + 3*10 = 50-x

9x+30 = 50-x

Add x to each side

9x+x+30 = 50-x+x

10x+30 = 50

Subtract 30 from each side

10x+30-30 = 50-30

10x= 20

Divide each side by 10

10x/10=20/10

x =2

Answer:

A direct variation equation to represent this ;  [tex]y = \frac{2}{3}x[/tex]

Step-by-step explanation:

Direct Variation states that a relationship between two variables in which one is a constant multiple of the other one.

In other words, when one of the variable changes then the other changes in proportion to the first.

If b is directly proportional to a, then the equation is in the i.e,

form [tex]b = ka[/tex] where k is the constant of variation.

Let y represents the number of muffins and x represents the amount of flour.

It is given that the number of muffins varies directly with the amount of flour you use.

As per the given statement:

y = 2 dozen corn and x = 3 cups of flour.

Then, by definition of Direct variation;

y = kx

Substitute the given values to find k;

[tex]2 = 3k[/tex]

Divide both sides by 3 we get;

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

then, equation is, [tex]y = \frac{2}{3}x[/tex]

Therefore, a direct variation equation to represent this situation is; [tex]y = \frac{2}{3}x[/tex]

Answer:

17.9

Step-by-step explanation:

Here we are given a right angled triangle with a known angle of 72°, length of the perpendicular to be 17 feet and we are to find the length of the hypotenuse x.

For that, we can use the formula for sin for which we need an angle and the lengths of base and hypotenuse.

[tex]sin \alpha =\frac{perpendicular}{x}[/tex]

So putting in the given values to get:

[tex]sin 72=\frac{17}{x} [/tex]

[tex]x=\frac{17}{sin 72}[/tex]

[tex]x=17.8[/tex]

Therefore, the length of six cars is the closest to 17.9.

Answer:

The probability is about 7.7%

Step-by-step explanation:

A probability is the ratio of the number of relevant outcomes and the number of all possible outcomes. To answer this question we do some counting first:

Consider two draws. We are interested in the second draw being an Ace, the first draw can either be an Ace or not, so, there are two cases of a relevant outcomes (all without replacement):

(number of relevant oucomes n ) = (number of cases of Ace-Ace draws) + (number of cases of NonAce-Ace draws):

[tex]n=4\cdot 3+{48\cdot4}= 204[/tex]

(number of possible outcomes) = (number of choices at first draw) * (number of choices at second draw) = 52 * 51 = 2652

The probability is then

P = 204/2652 = 0.0769, or about 7.7%

Answer:

The point [tex](\frac{\sqrt{3}}{2},-\frac{1}{2})[/tex]

Step-by-step explanation:

I added a graphic to the explanation.

Given the unit circle (the circle with radius equal to 1 unit centered at the point (0,0) ) we can represent its points only with an angle.

The point [tex]-(\frac{\pi}{6})[/tex] corresponds to the point that forms an angle of -30° respect to the positive axis-x (we measure the positive angles counterclockwise respect to the positive x-axis and the negative angles clockwise) because [tex]-(\frac{\pi}{6})[/tex] it is in radians and 180° = π radians ⇒

-(π/6) = - (180°/6) = - 30°

Given that we identify the point on the graph, we can find it coordinates using sine and cosine function :

[tex]sin(-30)=\frac{y1}{1} \\y1=-0.5=-\frac{1}{2}[/tex]

[tex]cos(-30)=\frac{x1}{1} \\x1=\frac{\sqrt{3}}{2}[/tex]

It is important to note that the hypotenuse of the right triangle which we used to apply sine and cosine is equal to 1 because is the radius of the unit circle.

The coordinates of the point are [tex](x1,y1)=(\frac{\sqrt{3}}{2},-\frac{1}{2})[/tex]

The point on the unit circle has the coordinates (√3/2, -1/2)

For a given angle a in an unit circle, the rectangular coordiantes of the point located in the circle are:

x = cos(a)

y = sin(a)

Here the angle is -(π/6), using the above relations, we will get the rectangular coordinates:

x = cos(-(π/6))

y = sin(-(π/6))

Simplify that:

x = √3/2

y = -1/2

The point is (√3/2, -1/2)

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Answer:

[tex]a=6[/tex], [tex]b=6\sqrt{2}[/tex], [tex]c=2\sqrt{3}[/tex], and [tex]d=6[/tex]

Step-by-step explanation:

Looking at the left triangle, we can solve for [tex]a[/tex] and [tex]c[/tex].

a:

[tex]a[/tex] is to the opposite side of 60° angle, also we know the hypotenuse, [tex]4\sqrt{3}[/tex]. The ratio that relates opposite with hypotenuse is SINE. Thus we can write:

[tex]sin(A)=\frac{opposite}{hypotenuse}\\sin(60)=\frac{a}{4\sqrt{3}}\\[/tex]

Cross multiplying and solving for [tex]a[/tex]:

[tex]sin(60)=\frac{a}{4\sqrt{3} }\\a=sin(60)*4\sqrt{3}\\a=\frac{\sqrt{3}}{2}*4\sqrt{3}\\a=\frac{12}{2}=6[/tex]

( we know [tex]sin(60)=\frac{\sqrt{3}}{2}[/tex] and also [tex]\sqrt{a}*\sqrt{a}=a[/tex] )

c:

[tex]c[/tex] is to the adjacent side of 60° angle, also we know the hypotenuse, [tex]4\sqrt{3}[/tex]. The ratio that relates adjacent with hypotenuse is COS. Thus we can write:

[tex]cos(A)=\frac{adjacent}{hypotenuse}\\cos(60)=\frac{c}{4\sqrt{3}}[/tex]

Cross multiplying and solving for [tex]c[/tex]:

[tex]cos(60)=\frac{c}{4\sqrt{3}}\\c=cos(60)*4\sqrt{3}\\c=\frac{1}{2}*4\sqrt{3}\\c=2\sqrt{3}[/tex]

( we know [tex]cos(60)=\frac{1}{2}[/tex] )

Looking at the triangle to the right, we can solve for [tex]b[/tex] and [tex]d[/tex].

b:

[tex]a[/tex] is to the opposite side of 45° angle. We have figured out that [tex]a=6[/tex]. Also we know that [tex]b[/tex] is the hypotenuse.The ratio that relates opposite with hypotenuse is SINE. Thus we can write:

[tex]sin(A)=\frac{opposite}{hypotenuse}\\sin(45)=\frac{6}{b}\\b=\frac{6}{sin(45)}\\b=\frac{6}{\frac{1}{\sqrt{2}}}\\b=6*\frac{\sqrt{2}}{1}\\b=6\sqrt{2}[/tex]

( we know [tex]sin(45)=\frac{1}{\sqrt{2}}[/tex] )

d:

To solve for [tex]d[/tex], we can use the pythagorean theorem. Given by:

[tex]a^2+b^2=c^2[/tex]

Where,

In the triangle on the right, [tex]b[/tex] is the hypotenuse and [tex]a[/tex] and [tex]d[/tex] are the two legs. Using pythagorean theorem and solving for [tex]d[/tex], we get:

[tex]d^2+a^2=b^2\\d^2+(6)^2=(6\sqrt{2})^2\\d^2+36=72\\d^2=72-36\\d^2=36\\d=6[/tex]

( we know that [tex]\sqrt{a}*\sqrt{a}=a[/tex] )

Looking at the answers, 2nd answer choice is right.

Answer:

Yes

Step-by-step explanation:

3+5 is 8 and 6+2 is also 8. You can draw 3 apples with 5 other apples, and count that there are 8 of them and draw 6 apples with 2 other apples that make up 8 apples. Or you can rearrange the apples to be 5 and 3 and 6 and 2.

Answer:

216 in squared

Step-by-step explanation:

using the formula, a cube has six sides so 6 squared is 36 times 6 equals to 216

Answer:

65550 is the population at the end of 2017

Step-by-step explanation:

Population at the beginning of 2106:  60000

Increased 15% in 2016

Increase = 60000*.15 = 9000

New population = 60000+9000 = 69000

The population at the beginning of 2017: 69000

Decrease by 5% in 2017

Decrease 69000*.05 = 69000*.05 = 3450

New population = 69000-3450=65550

The population at the end of 2017 is 65550

The function you're to evaluate is

... f(x) = 0.5x +3.40

Using function notation, the value for x = 2 is ...

... f(2)

For x = 3.5, the value is ...

... f(3.5)

and so on.

To actually evaluate the function, you need to put the value where x is in the function definition and do the arithmetic:

... f(2) = 0.5·2 + 3.40 = 1.00 +3.40 = 4.40

_____

The function gives the charge for a package based on its weight in ounces. The result of evaluating the function for the given weights is that you find the charges for delivery of those packages.

Answer:

D. differences

Step-by-step explanation:

One way to identify a linear function is by checking the differences over equal intervals. If they are the same, then the function is linear.

Answer:

D. Differences

Step-by-step explanation:

Linear function grow by equal differences over equal intervals as, on a graph, it is shown by a steady increasing line. Exponential functions, on the other hand, do not.

Answer:

C. The length of one lap.

Step-by-step explanation:

In the first four words, the problem statement tells you that L is the number of laps run. It is not the length of one lap.

_____

Comment on the problem presentation

Appropriate punctuation would be very helpful. Apparently, it is a 1/4-mile track, not a 14-mile track. Apparently, the distance is 1 1/2 miles, not 112 miles. Copying and pasting problem text often leaves out the special symbols used on some web sites. Some editing is usually needed.

Answer:

Congruent; Not Congruent; Congruent; Not Congruent; Not Congruent; Congruent

Step-by-step explanation:

We will use the distance formula to find the length of each segment:

[tex]AB=\sqrt{(-4--3)^2+(4-2)^2}=\sqrt{(-1)^2+(2)^2}=\sqrt{1+4}=\sqrt{5}\\\\BC=\sqrt{(-3--1)^2+(4-1)^2}=\sqrt{(-2)^2+(3)^2}=\sqrt{4+9}=\sqrt{13}\\\\AC=\sqrt{(2-1)^2+(-4--1)^2}=\sqrt{(1)^2+(-3)^2}=\sqrt{1+9}=\sqrt{10}[/tex]

In order for ABC to be congruent to DEF, AB must be congruent to DE, BC must be congruent to EF, and AC must be congruent to DF:

[tex]DE=\sqrt{(4-3)^2+(-2--4)^2}=\sqrt{(1)^2+(2)^2}=\sqrt{1+4}=\sqrt{5}\\\\EF=\sqrt{(3-1)^2+(-4--1)^2}=\sqrt{(2)^2+(-3)^2}=\sqrt{4+9}=\sqrt{13}\\\\DF=\sqrt{(4-1)^2+(-2--1)^2}=\sqrt{(3)^2+(-1)^2}=\sqrt{9+1}=\sqrt{10}[/tex]

Since AB is congruent to DE, BC is congruent to EF and AC is congruent to DF, the two triangles are congruent.

In order for ABC to be congruent to JKL, AB must be congruent to JK, BC must be congruent to KL, and AC must be congruent to JL.  We know the measurements of AB, BC and AC;

[tex]JK=\sqrt{(-4--2)^2+(-1--3)^2}=\sqrt{(-2)^2+(2)^2}=\sqrt{4+4}=\sqrt{8}\\\\KL=\sqrt{(-2--1)^2+(-3-0)^2}=\sqrt{(-1)^2+(-3)^2}=\sqrt{1+9}=\sqrt{10}\\\\JL=\sqrt{(-1--4)^2+(0--1)^2}=\sqrt{(3)^2+(1)^2}=\sqrt{9+1}=\sqrt{10}[/tex]

While AC is congruent to JL, the other two corresponding pairs of sides are not congruent.  Therefore the triangles are not congruent.

In order for ABC to be congruent to QRS, AB must be congruent to QR, BC must be congruent to RS, and AC must be congruent to QS.  We know the measurements of AB, BC and AC;

[tex]QR=\sqrt{(3-4)^2+(3-1)^2}=\sqrt{(-1)^2+(2)^2}=\sqrt{1+4}=\sqrt{5}\\\\RS=\sqrt{(3-1)^2+(3-0)^2}=\sqrt{(2)^2+(3)^2}=\sqrt{4+9}=\sqrt{13}\\\\QS=\sqrt{(4-1)^2+(1-0)^2}=\sqrt{(3)^2+(1)^2}=\sqrt{9+1}=\sqrt{10}[/tex]

Since AB is congruent to QR, BC is congruent to RS, and AC is congruent to QS, this means that the two triangles are congruent.

Since ABC is congruent to DEF, and ABC is not congruent to JKL, this means that triangle DEF is not congruent to triangle JKL.

Since ABC is congruent to QRS, and QRS is not congruent to JKL, this means that triangle QRS is not congruent to JKL.

Since ABC is congruent to DEF and ABC is congruent to QRS, this means that DEF is congruent to QRS by the transitive property.

Answer:

False

Step-by-step explanation:

It is the exponent of a base.

The variable in an exponential function is always the exponent of the power is false.

When the expression of function is such that it involves the input to be present as an exponent (power) of some constant, then such function is called exponential function.

Their usual form is specified below. They are written in several such equivalent forms.

The variable in an exponential function is always the exponent of the power is false. It is the exponent of a base.

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45° and 135°

If the smaller is represented by x, then the larger is 3x. The two angles are supplementary, so ...

... x + 3x = 180°

... 4x = 180°

... 180°/4 = x = 45°

... 3x = 135°

The two angles are 45° and 135°.

Answer:

45° and 135°

Step-by-step explanation:

If the smaller is represented by x, then the larger is 3x. The two angles are supplementary, so ...

... x + 3x = 180°

... 4x = 180°

... 180°/4 = x = 45°

... 3x = 135°

The two angles are 45° and 135°.

Step-by-step explanation:

Answer:

15

Step-by-step explanation: