-) Find the equation of the line that passes through (1,0) and (3,6).

Answer:

Δ ABC [tex]\bold{\sim}[/tex]  ΔPQR is similar.

Step-by-step explanation:

Similar triangles are two or more triangles that have the same shape, but their sides are in proportion.

For Question:

In Δ ABC and ΔPQR

AB:PQ =8:6=4:3

BC: QR=12:9=4:3

AC: PR=12:9=4:3

Since the length of any side of one triangle is by the corresponding side of another triangle, you will get the same number.

Therefore,

Δ ABC [tex]\bold{\sim}[/tex]  ΔPQR is similar.

Hence Proved:

Answer:

ΔABC ~ ΔPQR

Step-by-step explanation:

In similar triangles, corresponding sides are always in the same ratio.

Therefore, if triangle ABC is similar to triangle PQR then:

[tex]AB : PQ = BC : QR = AC : PR[/tex]

Substitute the side lengths into the ratio equation:

[tex]AB : PQ = BC : QR = AC : PR[/tex]

   [tex]8 : 6 \;\;\;\:= \;\;12 : 9 \;\;\;\:= \;\;12 : 9[/tex]

Simplify each ratio by dividing all parts of the ratio by their highest common factor:

[tex]\dfrac{8}{2}:\dfrac{6}{2}=\dfrac{12}{3}:\dfrac{9}{3}=\dfrac{12}{3}:\dfrac{9}{3}[/tex]

[tex]4:3=4:3=4:3[/tex]

As the corresponding sides of triangles ABC and PQR are in the same ratio, this proves that the two triangles are similar.

The original price of the wool suit was $795.

Let's denote the original price of the wool suit as "P."

We know that the suit was discounted by 20%, which means the discounted price is 80% (100% - 20%) of the original price.

Given that the discounted price is $636, we can set up the following equation:

0.8P = $636

To find the original price, we need to solve for P. We can do this by dividing both sides of the equation by 0.8:

P = $636 / 0.8

P = $795

Therefore, the original price of the wool suit was $795.

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The correct answer is option B: 10³ - 8y² + 3.

To obtain the equivalent expression, we can simplify the given expression by combining like terms. We have 107 as a constant term, -8y as the coefficient of the y term, and 3 as another constant term.

So, combining these terms, we get 10³ - 8y² + 3, which is equivalent to the given expression.

Option A is incorrect because it introduces an additional y term in the form of y², which is not present in the original expression.

Option C is incorrect because it only includes the y term and its coefficients, without considering the constant terms.

Option D is unrelated to the original expression as it introduces unrelated numbers and symbols.

Therefore, the correct equivalent expression is 10³ - 8y² + 3, as stated in option B.

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Answer:

25/9

Step-by-step explanation:

3 1/3 ÷ 1 1/5

3 1/3 = 10/3

1 1/5 = 6/5

10/3 ÷ 6/5 = 10/3 x 5/6 = 50/18 = 25/9

So, the answer is 25/9

The ratios are both identical (three fifths and three fifths). Option A

The ratios of sin x° and cos y° are given as;

sin θ = opposite/hypotenuse

cos θ = adjacent/hypotenuse

In the question given above, the value of adjacent is 5 and the value of perpendicular is 12.

By using Pythagoras theorem, we can find the value of the hypotenuse.

Hypotenuse² = Adjacent² + Perpendicular²

Hypotenuse² = 5² + 12²

Hypotenuse² = 25 + 144

Hypotenuse² = 169 = 13²

Hypotenuse = 13

The value of sin X° will be;

sin X =  5/13

cos Y = 5/13.

sin X° = cos Y° = 5/13

Therefore, both the ratios of sin x° and cos y° are equal and their value is 5/13.

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Complete question:

Use the image below to answer the following question. What relationship do the ratios of sin x° and cos y° share?

A right triangle is shown. The two angles that are not 90 degrees are marked x and y. The leg across from angle y measuring 4, another leg across from angle x measuring 3, and the hypotenuse measuring 5

The ratios are both identical (three fifths and three fifths).

The ratios are opposites (negative three fifths and three fifths).

The ratios are reciprocals (three fifths and five thirds).

The ratios are both negative (negative five thirds and negative three fifths).

Answer:

  x = 1

Step-by-step explanation:

You want to know the value of x in the trapezoid with adjacent angles (43x+2)° and 135°.

The two marked angles can be considered "consecutive interior angles" where a transversal crosses parallel lines. As such, they are supplementary.

  (43x +2)° +135° = 180°

  43x = 43 . . . . . . . . . . . . . . divide by °, subtract 137

  x = 1 . . . . . . . . . . . . . . . divide by 43

The value of x is 1.

__

Additional comment

Given that the figure is a trapezoid, we have to assume that the top and bottom horizontal lines are the parallel bases.

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To shift the graph left, we need to subtract 1 from x. This gives us F(x) = (x - 1)², which matches option D. Therefore, the correct answer is D.

The graph of F(x) is a parabolic graph, just like the graph of G(x)=x². F(x) is shifted to the left by 1 unit.

The equation for the graph of F(x) is F(x) = (x - 1)². This is because when a parabolic graph is shifted left or right, it affects the x-value of the vertex of the parabola.

The standard form of the equation for a parabolic graph is y = a(x - h)² + k, where (h, k) is the vertex of the graph and a determines whether the graph is facing up or down. In this case, the vertex of the graph of G(x)=x² is (0, 0).

When we shift this graph left by 1 unit, the vertex becomes (-1, 0). Therefore, the equation for F(x) is F(x) = a(x + 1)².

To determine the value of a, we can look at the y-intercept of the graph, which is (0, 1). Plugging these values into the equation, we get 1 = a(0 + 1)², which simplifies to 1 = a.

Therefore, the equation for F(x) is F(x) = (x + 1)². However, if we expand this equation, we get F(x) = x² + 2x + 1, which is not in the form given in the options.

To shift the graph left, we need to subtract 1 from x. This gives us F(x) = (x - 1)², which matches option D. Therefore, the correct answer is D.

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Answer:

The given triangles APSR and APQR are congruent by S.A.S. criteria.

Step-by-step explanation:

The S.A.S. (Side-Angle-Side) congruence criteria states that two triangles are congruent if their corresponding two sides and included angles are congruent to one another's corresponding two sides and included angle.

Here, we can observe that both triangles share the same side AP.

PR and PQ on the side are congruent.

- Both triangles share Angle P.

The supplied triangles APSR and APQR are therefore congruent according to S.A.S. requirements.

so we can conclude: S.A.S. for the triangles being congruent.

Answer:

$2100

Step-by-step explanation:

40% of 1500 is 600

1500 + 600 = 2100

The selling price will be $2100

When 'x' is 5, the Algebraic  expression 3x - 4 evaluates to 11.

To find the algebraic expression for "4 less than the product of 3 and some number," let's break it down step by step.

First, let's define the unknown number as 'x.' Since the problem states "the product of 3 and some number," we can express this as 3 * x or simply 3x.

Next, we want to subtract 4 from this product. The phrase "4 less than" indicates subtraction, so we subtract 4 from 3x. This can be represented as 3x - 4.

In summary, the algebraic expression for "4 less than the product of 3 and some number" is 3x - 4, where 'x' represents the unknown number.

To calculate the value of this expression for a specific value of 'x,' you substitute that value into the expression and simplify. For example, if 'x' is 5, you would substitute 5 into the expression:

3(5) - 4

This simplifies to:

15 - 4 = 11

So, when 'x' is 5, the expression 3x - 4 evaluates to 11.

In general, the algebraic expression allows us to represent a mathematical relationship between quantities using symbols and operations. By substituting specific values into the expression, we can calculate the corresponding results.

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Note the complete question:

Algebraic expression for 4 less than the product of 3 and some number

What is the algebraic expression for "4 less than the product of 3 and some number?

Please help me understand the meaning of how to calculate.

The values of w, x, y, z are 120°, 60°, 120° , 60° respectively.

Angles in parallel lines are angles that are created when two parallel lines are intersected by another line called a transversal.

lines PQ and RS are parallel to each other and AB and CD are also parallel to each other. A pair of parallel line will serve as transversal.

Angles on parallel lines can be ;

Vertically opposite, alternate, corresponding and in each case the angles are equal.

w = 120°( vertically opposite angles)

x = 180-120 = 60( corresponding angles)

x = z = 60° ( corresponding angles)

y = 180-60 = 120°( angles on a straight line)

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This is the same as writing y = sqrt(4(x+5)) - 1

===============================================

Explanation:

The given graph appears to be a square root function.

The marked points on the curve are:

Reflect those points over the line y = x. This will have us swap the x and y coordinates.

Recall the process of reflecting over y = x means we're looking at the inverse. The inverse of a square root function is a quadratic.

----------

Let's find the quadratic curve that passes through (1,-4), (3,-1) and (5,4).

Plug the coordinates of each point into the template y = ax^2+bx+c.

For instance, plug in x = 1 and y = -4 to get...

y = ax^2+bx+c

-4 = a*1^2+b*1+c

-4 = a+b+c

Do the same for (3,-1) and you should get the equation -1 = 9a+3b+c

Repeat for (5,4) and you should get 4 = 25a+5b+c

We have this system of equations

Use substitution, elimination, or a matrix to solve that system. I'll skip steps, but you should get (a,b,c) = (1/4, 1/2, -19/4) as the solution to that system.

In other words

a = 1/4, b = 1/2, c = -19/4

We go from y = ax^2+bx+c to y = (1/4)x^2+(1/2)x-19/4

----------

Next we complete the square

y = (1/4)x^2+(1/2)x-19/4

y = (1/4)( x^2+2x )-19/4

y = (1/4)( x^2+2x+0 )-19/4

y = (1/4)( x^2+2x+1-1 )-19/4

y = (1/4)( (x^2+2x+1)-1 )-19/4

y = (1/4)( (x+1)^2-1 )-19/4

y = (1/4)(x+1)^2- 1/4 - 19/4

y = (1/4)(x+1)^2 + (-1-19)/4

y = (1/4)(x+1)^2 - 20/4

y = (1/4)(x+1)^2 - 5

The equation is in vertex form with (-1,-5) as the vertex. It's the lowest point on this parabola. Placing it into vertex form allows us to find the inverse fairly quickly.

----------

The last batch of steps is to find the inverse.

Swap x and y. Then solve for y.

y = (1/4)(x+1)^2 - 5

x = (1/4)(y+1)^2 - 5

x+5 = (1/4)(y+1)^2

(1/4)(y+1)^2 = x+5

(y+1)^2 = 4(x+5)

y+1 = sqrt(4(x+5))

y = sqrt(4(x+5)) - 1

I'll let the student check each point to confirm they are on the curve y = sqrt(4(x+5)) - 1.

You can also use a tool like GeoGebra to verify the answer.

Answer:

(9,16)

Step-by-step explanation:

We Know

The equation y = 3x - 2

Find the y coordinate (9, ?)

We just simply put 9 in for x and solve for y

y = 3(9) - 2

y = 18 - 2

y = 16

So, the coordinate are (9,16)

The scaled triangle will be larger than the initial size by a factor 2.

The scaled square will be smaller than the initial side by a factor 4

Dilation refers to a transformation that changes the size of a geometric figure without altering its shape.

Dilation involves scaling an object by a certain factor, that might result in  enlarging or reducing its dimensions uniformly in all directions.

Based on the given diagram, the new length and size of the object is calculated as follows;

For the triangle, (measure the length with ruler)

For the square; (measure the length with ruler)

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Answer:

Kirsten makes $12.50 per hour

Step-by-step explanation:

The total is $432.50

$5 is taken out of her paycheck, so that will be negative.

35 hours is the time she spent working.

The unknown variable we are looking for is the amount she makes in one hour.

Setting up the equation, we get:

$432.50 = $35x - $5

where x is the amount she makes every hour

Solve for x:

$437.50=$35x

$437.50 / $35 = x

Therefore, x = $12.50

Therefore, the coordinates of point B are (-5.5, -10).

To find the coordinates of point B, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint between two points (x1, y1) and (x2, y2) are given by:

Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

We are given the midpoint coordinates as (-3, -5) and the coordinates of point A as (-0.5, 0). Let's assume the coordinates of point B as (x, y). Plugging the known values into the midpoint formula, we have:

(-3, -5) = ((-0.5 + x) / 2, (0 + y) / 2)

Simplifying, we get:

-3 = (-0.5 + x) / 2 ... (1)

-5 = y / 2 ... (2)

From equation (2), we can solve for y:

y = -5 * 2

y = -10

Now, substituting this value of y in equation (1), we can solve for x:

-3 = (-0.5 + x) / 2

-6 = -0.5 + x

x = -6 + 0.5

x = -5.5

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The Lagrange interpolating polynomial of degree 1 is P₁(x) = (f₁ - fo)x + (foxo - f₁xo) / (x₁ - xo).

To derive the Lagrange interpolating polynomial of degree 1, let's consider two distinct points: (x₁, f₁) and (xo, fo). The Lagrange interpolating polynomial of degree 1 can be expressed as:

P₁(x) = L₁(x)f₁ + L₂(x)fo,

where L₁(x) and L₂(x) are the Lagrange basis polynomials defined as:

L₁(x) = (x - xo) / (x₁ - xo),

L₂(x) = (x - x₁) / (xo - x₁).

Substituting these expressions into P₁(x), we have:

P₁(x) = [(x - xo) / (x₁ - xo)]f₁ + [(x - x₁) / (xo - x₁)]fo.

Simplifying further:

P₁(x) = [f₁(x - xo) + fo(x - x₁)] / (x₁ - xo).

Expanding the numerator:

P₁(x) = [f₁x - f₁xo + fox - fox₁] / (x₁ - xo).

Finally, we can rearrange the terms to obtain the Lagrange interpolating polynomial of degree 1:

P₁(x) = (f₁ - fo)x + (foxo - f₁xo) / (x₁ - xo).

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Answer: The expression, 3v^4, means that the variable, v, is multiplied by itself 4 times (v * v * v * v) and then is multiplied after by the coefficient 3.

Step-by-step explanation:

Exponent indicates the number of times a number needs to be multiplied by itself.

For example if your given x^2

that simply means that the variable, x, is being multiplied by itself twice:

x * x

if your given 2z^9

that means that the variable, z, is first multiplied by itself nine times and after that it is multiplied by the coefficient, two.

The equations that represent a line parallel to 3x - 4y = 7 and passes through the point (-4, -2) are:

0 = 3x - 4y - 4

3x - 4y = 4

To determine which equations represent a line that is parallel to the given line and passes through the point (-4, -2), we need to find the equations that have the same slope as the given line.

The equation 3x - 4y = 7 can be rewritten in slope-intercept form as y = (3/4)x - 7/4. From this form, we can see that the slope of the given line is 3/4.

Now let's analyze the options:

Oy = -3x + 1: This equation has a slope of -3, which is not equal to the slope of the given line (3/4). Therefore, this option does not represent a line parallel to the given line.

0 = 3x - 4y - 4: This equation can be rewritten as 3x - 4y = 4. Comparing this to the given line, we can see that it has the same coefficients of x and y, which means it has the same slope of 3/4. Therefore, this option represents a line parallel to the given line.

4x - 3y = -3: This equation has a slope of 4/3, which is not equal to the slope of the given line (3/4). Therefore, this option does not represent a line parallel to the given line.

Oy - 2 = -(x - 4): This equation can be rewritten as y = -x + 2. The slope of this line is -1, which is not equal to the slope of the given line (3/4). Therefore, this option does not represent a line parallel to the given line.

Oy + 2 = 2(x + 4): This equation can be rewritten as y = 2x + 6. The slope of this line is 2, which is not equal to the slope of the given line (3/4). Therefore, this option does not represent a line parallel to the given line.

In conclusion, the equations that represent a line parallel to 3x - 4y = 7 and passes through the point (-4, -2) are:

0 = 3x - 4y - 4

3x - 4y = 4

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The length of CF, considering the Pythagorean Theorem in this problem, is given as follows:

CF = 26.08 m.

The Pythagorean Theorem states that in the case of a right triangle, the square of the length of the hypotenuse, which is the longest side,  is equals to the sum of the squares of the lengths of the other two sides.

Hence the equation for the theorem is given as follows:

c² = a² + b².

In which:

The length of EC is given as follows:

(EC)² = 10² + 24²

[tex]EC = \sqrt{10^2 + 24^2}[/tex]

EC = 26.

Then the length of CF is given as follows:

CF² = 26² + 2²

[tex]CF = \sqrt{26^2 + 2^2}[/tex]

CF = 26.08 m.

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[tex]\underline{\underline{\purple{\huge\sf || ꪖꪀᦓ᭙ꫀ᥅}}}[/tex]

First, we can simplify each expression:

(10⁰x10¹x1¹⁰) = 10¹¹

(10x10¹) = 100

(10⁰+10¹+1¹⁰) = 1 + 10 + 1,000,000,000 = 1,000,000,011

(10⁰+10¹x1¹⁰) = 10¹⁰

Now we can arrange them in increasing order:

10¹¹ < 10¹⁰ < 100 < 1,000,000,011

So the correct order from smallest to largest is:

(10⁰x10¹x1¹⁰) (10⁰+10¹x1¹⁰) (10 x 10¹) (10⁰+10¹+1¹⁰)

or

(10¹¹) (10¹⁰) (100) (1,000,000,011)

The test score that is an outlier is 60. Option A

A data point that dramatically deviates from the dataset's main pattern or trend is referred to as an outlier.

From the information given, we have that;

The score are;

60, 69, 79, 80, 86, 86, 86, 89, 90, and 100.

We can see that the outlier is 60.

This is due to the fact that all of the other scores, which range from 69 to 100, are considerably higher and closer in value.

The score of 60, however, is considerably lower than the others pupils.

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To simplify the expression 14√27 - √3, we can simplify the radicals and then perform the subtraction:

First, let's simplify the radicals:

√27 = √(9 × 3) = √9 × √3 = 3√3

Now we can rewrite the expression:

14√27 - √3 = 14(3√3) - √3

Next, distribute the 14:

14(3√3) - √3 = 42√3 - √3

Finally, combine like terms:

42√3 - √3 = 41√3

So the simplified expression is 41√3.

Answer:

7: 63

8: 73

arithmetic sequence

y = 10x - 7

or f(n) = 10x -7

or

[tex]a_{n}[/tex] = 3 + (n-1)10

Step-by-step explanation:

the output increases by 10 every time that the input increases by 1.  That gives us our common difference or slope.  The y intercept is -7. That is the value is you worked backwards until you get to n = 0.  The initial value  is 3.  That is when n is 1.

When n is 3, f(n) is 23

When n is 2, f(n) is 13

When n is 1, f(n) is 3

When n is 0, f(n) is -7

I am not sure if this is clear.  I am assuming that you have a lot of knowledge of linear equations and how to write arithmetic sequence.  If my explanation is confusing it is me and not you.

Answer:

a. 63,73

b. Arithmetic sequence

c.t(n)=10n-7

Explanation:

a. Here is the completed table:

n | t(n)

4 | 33

5 | 43

6 | 53

7 | 63

8 | 73

b.

The type of sequence is arithmetic.

An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant.

In this case, the difference between any two consecutive terms is 10.

c.

The equation for the arithmetic sequence is:

t(n)=a+(n-1)d

where:

For Question:

Now

equation becomes:

t(4) = a+(4-1)10

33=a+30

a=33-30

a=3

Now, the Equation becomes

t(n) = 3+(n-1)10

t(n) = 3+10n-10

t(n)=10n-7

Therefore, an appropriate confidence interval for μ30−μ60μ30​−μ60​ (the difference in mean body temperature) would be −0.796−0.796 to −0.404−0.404 (rounded to the appropriate decimal places).

To estimate the difference in the mean body temperature after exercising for 30 minutes versus 60 minutes, we can construct a confidence interval. Since the conditions for inference have been assumed to be met, we can proceed with the following steps:

   Calculate the difference in sample means: xˉ30−xˉ60=38.3−38.9=−0.6xˉ30​−xˉ60​=38.3−38.9=−0.6.

   Determine the standard error of the difference, which is given by the formula:

   SE=(s302n30)+(s602n60)SE=(n30​s302​​)+(n60​s602​​)

Substituting the values, we have:

SE=(0.27218)+(0.29224)≈0.100SE=(180.272​)+(240.292​)  ​≈0.100.

   Choose an appropriate confidence level. Let's say we want a 95% confidence interval.

   Using the standard error, calculate the margin of error, which is given by multiplying the critical value (corresponding to the chosen confidence level) by the standard error. For a 95% confidence level, the critical value is approximately 1.96.

   Margin of Error = 1.96 * 0.100 ≈ 0.196.

   Construct the confidence interval by subtracting and adding the margin of error to the difference in sample means:

   −0.6−0.196−0.6−0.196 to −0.6+0.196−0.6+0.196.

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The value of x in the expression given is -7/6

(6x + 9)/2 =

First step is to cross multiply

6x + 9 = 2

Collect like terms

6x = 2 - 9

6x = -7

divide both sides by 6 to isolate x

x = -7/6

Therefore the value of x in the expression given is -7/6

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As the given equation is (36)ˣ⁺²  = 6 , then the solution for the given equation is x ≈ -1.54.

The given equation in the question is (36)ˣ⁺²  = 6

according to the given equation , the goal is to solve the equation for x.

For solving this equation ,

we'll have to use logarithmic operations which is as follows:  

log36(6) = x+2.

We will be further using a calculator to determine the logarithmic value: log36(6) ≈ 0.4607.

Now, By subtracting 2 from both sides, we will get the value of x as:

x ≈ -1.54.

When we round off this value to the nearest 100th, we will get  -1.54.

Thus , the solution for the given equation is x ≈ -1.54.

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The algebra class of 14 students in the Algebra 2 class of 23 students do not play basketball or baseball.

The number of students in a certain Algebra 2 class of 23 students who play basketball and baseball is 9, given that 7 of them play basketball and 2 of them play baseball.

The number of students in the class who do not play basketball or baseball is given by the number of students who do not play basketball plus the number of students who do not play baseball and the students who do not play basketball or baseball.

However, since each student is either playing basketball or baseball or neither, we can say that the total number of students who do not play basketball or baseball is given by the number of students in the class minus the total number of students who play basketball or baseball.

Thus, the number of students who do not play basketball or baseball is given by:

23 - 9 = 14

Therefore, 14 students in the Algebra 2 class of 23 students do not play basketball or baseball.

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