(x^2-6x+9)^2-15(x^2-6x+10)=1

Sariah has just begun training for a half-marathon, which is latex: 13.1 13 . 1 miles. since she was on vacation, she started the training program later than the rest of her running club. there are latex: 6 6 weeks of training runs remaining before the race. in her first week of training, sariah ran latex: 3 3 miles. she ran latex: 4.5 4 . 5 miles the second week and latex: 6 6 miles the third week. if she continues to increase the length of her runs the same way, will there be enough time left in the training program for her to get up to half-marathon distance? describe how you would solve this problem using polya's four-step problem-solving method. complete each task as part of your response. be sure to number your answers task 1–task 4 so that your instructor can tell which part you are answering. otherwise, you may not receive credit. Understand the problem.

Task 1: Read the problem and restate in y0ur own words what the question asks.

2. Formulat3 a plan.

Task 2: Identify the model y0u would use to represent the problem and explain why y0u chose that model.

3. Implement th3 plan.

Task 3: Solve the problem and state y0ur answer.

4. Review th3 results.

Task 4: Explain the problem-solving process and how y0u know y0ur answer is correct.

Half of a marathon's distance, or 21.0975 kilometers (13 miles 192.5 yards), is covered during a half marathon. Using nearly the same course with a later start, an earlier finish, or shortcuts, a half marathon event is sometimes held alongside a marathon or a 5K race. If finisher medals are given out, they might not look exactly like the ones given out for the full marathon. Although these measurements are rounded and not officially correct, the half marathon is frequently referred to as a 21K, 21.1K, or 13.1 miles.

The International Association of Athletics Federations recognizes a half marathon world record.

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The area of the shaded sector is 144π units squared.

To find the area of the shaded sector:

Given - The central angle of the sector is, θ [tex]=\frac{8\pi }{9} rad[/tex].

The radius of the circle is, [tex]R=18 units[/tex].

We know that the area of a sector of a circle of radius 'R' and central angle θ is given as:

[tex]A=\frac{1}{2} R^{2}[/tex]θ

Insert, θ [tex]=\frac{8\pi }{9} ,R=18[/tex] and obtain:

[tex]A=\frac{1}{2} *18^{2} *\frac{8\pi }{9} \\A=\frac{(324*4)}{9} \pi \\A=(36*4)\pi \\A=144\pi units^{2}[/tex]

Therefore, the area of the shaded sector is 144π units squared.

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The complete question is given below:

The measure of central angle QRS is StartFraction 8 pi Over 9 EndFraction radians. What is the area of the shaded sector? 36Pi units squared 72Pi units squared 144Pi units squared 324Pi units squared

It will cost $375 to hire the doctor for 45 minutes

The doctor charges $500 for an hour

60 minutes makes 1 hour

60 minutes= $500

Therefore the amount that will be charged for 45 minutes can be calculated as follows

60 minutes= $500

45 minutes= x

Cross multiply both sides

60x= 500×45

60x= 22500

x= 22500/60

x= 375

Hence it will cost $375 to hire the doctor for 45 minutes

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Answer:

23.  216

24.  56

25.  52

26.  -7

27.  5

Step-by-step explanation:

Work out the differences between the terms until the differences are the same:

[tex]1 \underset{+7}{\longrightarrow} 8 \underset{+19}{\longrightarrow} 27 \underset{+37}{\longrightarrow} 64 \underset{+61}{\longrightarrow} 125[/tex]

    [tex]7 \underset{+12}{\longrightarrow} 19 \underset{+18}{\longrightarrow} 37 \underset{+24}{\longrightarrow} 61[/tex]

        [tex]12 \underset{+6}{\longrightarrow} 18 \underset{+6}{\longrightarrow} 24[/tex]

As the third differences are the same, the sequence is cubic and will contain an n³ term.  The coefficient of n³ is always a sixth of the third difference.  Therefore, the coefficient of n³ = 1.

To work out the nth term of the sequence, write out the numbers in the sequence n³ and compare this sequence with the sequence in the question.

[tex]\begin{array}{| c | c | c | c | c | c |}\cline{1-6} n&1 & 2 & 3&4&5 \\\cline{1-6} n^3 & 1 & 8 & 27 & 64 & 125 \\\cline{1-6} \sf sequence & 1 & 8 & 27 & 64 & 125 \\\cline{1-6}\end{array}[/tex]

Therefore, the nth term is:

[tex]a_n=n^3[/tex]

So the next term in the sequence is:

[tex]\implies a_6=6^3=216[/tex]

Work out the differences between the terms until the differences are the same:

[tex]2 \underset{+4}{\longrightarrow} 6 \underset{+6}{\longrightarrow} 12 \underset{+8}{\longrightarrow} 20 \underset{+10}{\longrightarrow} 30 \underset{+12}{\longrightarrow} 42[/tex]

     [tex]4 \underset{+2}{\longrightarrow} 6 \underset{+2}{\longrightarrow} 8 \underset{+2}{\longrightarrow} 10 \underset{+2}{\longrightarrow} 12[/tex]

As the second differences are the same, the sequence is quadratic and will contain an n² term.  The coefficient of n² is always half of the second difference.  Therefore, the coefficient of n² = 1.

To work out the nth term of the sequence, write out the numbers in the sequence n² and compare this sequence with the sequence in the question.

[tex]\begin{array}{| c | c | c | c | c | c | c |}\cline{1-7} n&1 & 2 & 3&4&5&6 \\\cline{1-7} n^2 & 1 & 4 & 9 & 16 & 25 &36\\\cline{1-7} \sf operation & +1 & +2 & +3 & +4 & +5 & +6 \\\cline{1-7} \sf sequence & 2 & 6 & 12 & 20 & 30 & 42 \\\cline{1-7}\end{array}[/tex]

Therefore, the nth term is:

[tex]a_n=n^2+n[/tex]

So the next term in the sequence is:

[tex]\implies a_7=7^2+7=56[/tex]

Work out the differences between the terms until the differences are the same:

[tex]4 \underset{+3}{\longrightarrow} 7 \underset{+5}{\longrightarrow} 12 \underset{+7}{\longrightarrow} 19 \underset{+9}{\longrightarrow} 28 \underset{+11}{\longrightarrow} 39[/tex]

     [tex]3 \underset{+2}{\longrightarrow} 5 \underset{+2}{\longrightarrow} 7 \underset{+2}{\longrightarrow} 9 \underset{+2}{\longrightarrow} 11[/tex]

As the second differences are the same, the sequence is quadratic and will contain an n² term.  The coefficient of n² is always half of the second difference.  Therefore, the coefficient of n² = 1.

To work out the nth term of the sequence, write out the numbers in the sequence n² and compare this sequence with the sequence in the question.

[tex]\begin{array}{| c | c | c | c | c | c | c |}\cline{1-7} n&1 & 2 & 3&4&5&6 \\\cline{1-7} n^2 & 1 & 4 & 9 & 16 & 25 &36\\\cline{1-7} \sf operation & +3 & +3 & +3 & +3 & +3 & +3 \\\cline{1-7} \sf sequence & 4&7&12&19&28&39 \\\cline{1-7}\end{array}[/tex]

Therefore, the nth term is:

[tex]a_n=n^2+3[/tex]

So the next term in the sequence is:

[tex]\implies a_7=7^2+3=52[/tex]

Given numbers:

-1, 2, -3, 4, -5, 6

As the list of numbers does not increase or decrease, we cannot apply the same method as the previous questions.

[tex]\begin{array}{| c | c | c | c | c | c | c |}\cline{1-7} n&1 & 2 & 3&4&5&6 \\\cline{1-7} \sf list & -1 & 2 & -3 & 4 & -5 & 6 \\\cline{1-7}\end{array}[/tex]

From comparing the position of the term to its number in the list, we can see that the nth term is the same as n, but the odd numbers are negative and the even numbers are positive.

Therefore, the rule is:

[tex]\implies a_n=n \quad \textsf{where }n \textsf{ is an even natural number}[/tex]

[tex]\implies a_n=-n \quad \textsf{where }n \textsf{ is an odd natural number}[/tex]

So the next term in the sequence is:

[tex]\implies a_7=-7[/tex]

(since 7 is an odd natural number)

Given numbers:

5, 3, 5, 5, 3, 5, 5, 5, 3, 5, 5, 5, 5, 3, 5, 5, 5, 5

The pattern of these numbers appears to be an ascending number of 5s with a 3 in between:

Therefore, the next number of 5s will be five.  This means the next number in the list will be 5, since there are only four 5s after the last 3.

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Answer: 50

Step-by-step explanation:

50% of the class passed the first test, so that 25% that passed both test has to be part of the 50% that passed the first test. 25% that passed both is 50% of the 50% that passed the first test. Hopefully that makes sense.

Answer:

=> Equilateral Triangle

Step-by-step explanation:

Equilateral Triangle is having 2-d shape with all equal sides and the sum of angles of any triangle is 180° .

Answer:

Step-by-step explanatio

By the ratio test, the series converges for all [tex]x[/tex], since

[tex]\displaystyle \lim_{k\to\infty} \left|\frac{(k+1)^2 x^{k+4}}{(k+1)!} \cdot \frac{k!}{k^2 x^{k+3}}\right| = |x| \lim_{k\to\infty} \frac{(k+1)^2 k!}{k^2 (k+1)!} = |x| \lim_{k\to\infty} \frac1{k+1} = 0 < 1[/tex]

Then the radius of convergence is R = ∞, and the interval of convergence is the entire real line, -∞ < x < ∞.

The radius of convergence R is ∞ and the interval of convergence is (-∞, ∞) for the given power series. This can be obtained by using ratio test.  

Ratio test is the test that is used to find the convergence of the given power series.  

First aₙ is noted and then aₙ₊₁ is noted.

For  ∑ aₙ,  aₙ and aₙ₊₁ is noted.

[tex]\lim_{n \to \infty} | \frac{a_{n+1} }{a_{n} } |[/tex] = β

Here aₙ = (n²/n!) xⁿ⁺³  and  aₙ₊₁ = ((n+1)²/(n+1)!) xⁿ⁺¹⁺³ = ((n+1)²/(n+1)!) xⁿ⁺⁴

Now limit is taken,

[tex]\lim_{n \to \infty} | \frac{a_{n+1} }{a_{n} } |[/tex] = [tex]\lim_{n \to \infty} | \frac{((n+1)^{2} /(n+1)!) x^{n+4} }{(n^{2} /n!) x^{n+3} } |[/tex]

                      = [tex]\lim_{n \to \infty} | \frac{((n+1)^{2} ) n!x^{n+4} }{(n+1)!(n^{2} ) x^{n+3} } |[/tex]

                      = [tex]\lim_{n \to \infty} | \frac{((1+1/n)^{2} ) x }{(n+1)} |[/tex]  

                      =   [tex]\lim_{n \to \infty} | \frac{ x }{(n+1)} |[/tex]  = 0 < 1

Since the limit is less than 1 the series is converging.

We get that,

interval of convergence = (-∞, ∞)

radius of convergence R = ∞

Hence the radius of convergence R is ∞ and the interval of convergence is (-∞, ∞) for the given power series.

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Answer:

(2 dimes + 6 pennies)

(4 nickels + 6 pennies)

(1 dime + 2 nickels + 6 pennies)

( 2 dimes + 1 nickels + 1 penny)

(1 dime + 3 nickels + 1 penny)

5 different ways that Jamie can make 26 cents.

By the comparison test, the series converges.

We have

[tex]\dfrac1{k\sqrt{k+2}} \le \dfrac1{k \sqrt k} = \dfrac1{k^{3/2}}[/tex]

so we can compare to a convergent [tex]p[/tex]-series,

[tex]\displaystyle \sum_{k=1}^\infty \frac1{k\sqrt{k+2}} \le \sum_{k=1}^\infty \frac1{k^{3/2}} < \infty[/tex]

By the comparison test, the series converges.

What are convergence series and diverging series?

If the infinite series converges to a real number it is called converging if not then it is called diverging series.

If the larger series is convergent the smaller series must also be convergent. Likewise, if the smaller series is divergent then the larger series must also be divergent.

For this series to be converging it must follow the following :

⇒ [tex]| \frac{t_{n+1} }{t_{n} } | \leq 1[/tex]

⇒Putting n = 1

[tex]| \frac{t_{2} }{t_{1} } | \leq 1[/tex]

[tex]\t_{2} t_{1}[/tex] [tex]t_{2}[/tex]= 1/4

[tex]t_{1}[/tex]=1/[tex]\sqrt{2}[/tex]

[tex]\frac{t_{2} }{{t_{1} }}[/tex] = 1/2[tex]\sqrt{2[/tex]

⇒ t₂/t₁≤1

⇒ Hence it converges

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The minimum and maximum values are (16/9)√3  and -(16/9)√3.

According to the statement

we have given that the function

First of all, since the constraint's graph is a circle, which is a closed loop, and f is continuous in [tex]R^2[/tex], there must exist a constrained global maximum and minimum value.

Lagrange Multipliers:

f(x,y) = xy^2

g(x,y) = x^2 + y^2

We want ∇f = λ∇g so we get the following system of equations

1. y^2 = 2λx

2. 2xy = 2λy

3. x^2 + y^2 = 4 ← The constraint.

Equation 2 implies that y = 0 or λ = x

We can ignore y = 0, since that will make f(x,y) = 0 and clearly f(x,y) takes on both positive and negative values subject to the constraint.

Plugging in the alternative, λ = x to equation 1 gives y^2 = 2x^2.

Plugging this into the constraint gives 3x^2 = 4 so that x^2 = 4/3 and y^2 = 8/3

Taking square roots gives

x = ±√(4/3) = ±(2/3)√3

y = ±√(8/3) = ±(2/3)√6

At the points < (2/3)√3 , ±(2/3)√6 >, f(x,y) = (16/9)√3 ← Maximum

At the points < -(2/3)√3 , ±(2/3)√6 >, f(x,y) = -(16/9)√3 ← Minimum

So, The minimum and maximum values are (16/9)√3  and -(16/9)√3.

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The true statement is (c) Amal's work is incorrect because it includes a positive two instead of a negative two in the answer

The complete question is added as an attachment

From the table, we have

Divisor = x^2 - 3x + 4

Quotient = 2x + 5

Dividend = -2x^3 + 11x^2 - 23x + 20

See that the signs of the leading coefficients of the dividend and the divisor are different

This means that the leading coefficient of the quotient must be negative

From the question, we have:

Quotient = 2x + 5

The expression 2x + 5 has a positive leading coefficient

Hence, the true statement is (c) Amal's work is incorrect because it includes a positive two instead of a negative two in the answer

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Solving a linear equation, we conclude that the book has 276 pages in total.

From pages 1 to 9, each page needs one digit, so at this point we have 9 digits.

From page 10 to 99, each one needs 2 digits. Then in these pages we have:

(99 - 9)*2 = 180 digits.

So at this point we have 180 + 9 = 189 pages.

Now, for each page after this point, we will add 3 digits more.

So if there are x pages after this point, the total number of digits will be:;

N = 189 + x*3

Now we need to solve this linear equation for N = 720 digits, then:

720 = 189+ x*3

720 - 189 = x*3

531 = x*3

531/3 = x = 177

So there are 177 pages after page number 99, this means that the book has:

99 + 177 = 276 pages in total.

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The value of x from the given diagram is 20

According to theorem, the right triangle altitude theorem is a result in elementary geometry that describes a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse.

Using the theorem above;

RT^2 = 9 * 16

RT^2 = 144

R = 12 units

Determine the value of x using the Pythagoras theorem;

x² =12² + 16²

x² = 144 + 256

x² = 400

Take the square root of both sides

x = √400

x = 20

Hence the value of x from the given diagram is 20

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Answer:

x = 3

Step-by-step explanation:

Given the information from the question, we can deduce that:

3 - x = x - 3

Now our goal here is to find x.

3 - x = x -3

3 - x - 3  = x - 3 - 3

- x = x - 6

- x - x = x - 6 - x

- 2x = - 6

x = [tex]\frac{-6}{-2}[/tex] = 3

Answer: 3

Step-by-step explanation: The only value for x in which 3 - x is the same as x - 3 is where x does not affect the value of 3 whatsoever. We can also set this up algebraically to find this out algebraically.

x - 3 = 3 - x

Adding 3 to both sides, we have

x = 6 - x

Adding x to both sides we have

2x = 6

Dividing by 2 from both sides, we get

x = 3

Step-by-step explanation:

i donot know

sorry about this

This might help a bit, i hope

Order (B) 5 × 6 of the matrix can be multiplied by matrix a to create matrix ab.

To find the order of matrix:

To prove:

Therefore, order (B) 5 × 6 of the matrix can be multiplied by matrix a to create matrix ab.

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The standard deviation exists as the positive square root of the variance.

So, the standard deviation = 6.819.

Given data set: 15, 17, 23, 5, 21, 19, 26, 4, 14

To calculate the mean of the data.

We know that mean exists as the average of the data values and exists estimated as:

Mean [tex]$=\frac{15+17+23+5+21+19+26+4+14}{9}[/tex]

Mean [tex]$=\frac{144}{9}[/tex]

Mean = 16

To estimate the difference of each data point from the mean as:

Deviation:

15 - 16 = -1

17 - 16 = 1

23 - 16 = 7

5 - 16 = -11

21 - 16 = 5

19 - 16 = 3

26 - 16 = 10

4 - 16 = -12

14 - 16 = -2

Now we have to square the above deviations we obtain:

1 , 1, 14, 121, 25, 9, 100, 144, 4

To estimate the variance of the above sets:

variance [tex]$=\frac{1+ 1+14+ 121+25+ 9+ 100+ 144+ 4}{9}[/tex]

Variance [tex]$=\frac{419}{9}[/tex]

Variance = 46.5

The standard deviation exists as the positive square root of the variance.

so, the standard deviation [tex]$\sqrt{46.5} =6.819[/tex] .

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The figure 980 inches can be written as 980.0 inches. It contains two non-zero digits and a zero trailing before the decimal point. Hence, it has three significant figures.

Rules to Determine Number of Significant Figures:

There are three fundamental rules that one can use to determine the significant figures in any given number or value. These rules go as follows:

For the portion followed by a decimal point, zeroes at the end are not counted, but zero between two non-zero digits and zero in the initial portion of the floating point are counted while taking note of the significant figures in a decimal value.

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The length of rectangle is 25 feet and the width  of rectangle is 135 feet.

According to the question,

The perimeter of a rectangle is 320 feet. The length and width if the length is an odd integer and the width is 5 times the next consecutive odd integer.

Odd integers that follow each other and they differ by 2. If x is an odd integer, then x + 2, x + 4 and x + 6 are consecutive odd integers. The perimeter of a rectangle is 320 feet. Let the length of a rectangle is x. The next consecutive odd integer is x+2.width = 5(x+2)

    =5x+10.

Let, Length - l:Width - w; l = 5w and the perimeter of a rectangle is 320 feet. Formula for perimeter of rectangle is 2(Length + Breadth). Perimeter of a rectangle is the sum of the length of all sides of the rectangle. The perimeter of a rectangle formula is,

P = 2(l + b)

In words, it is equal to the sum of two times the length and two times the breadth of the rectangle. Perimeter for any figure is defined as the length of its boundary.

2(length + width) = 320

Length+ width=160

x+5x+10=160

6x=150

x=25

Length = 25 feet

width = 5(25+2)=135 feet.

Hence, the length of rectangle is 25 feet and the width of rectangle is 135 feet..

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Answer:

1/4

Step-by-step explanation:

To transform PQR into P'Q'R, dilate the preimage by 1/4, or shrink it by a scale factor of 4 because 3/12 = 1/4

Answer:

(0, -5/2).

Step-by-step explanation:

The coordinates of the mid point of (x1, y1) and (x2, y2) are

(x 1 + x2)/2 , (y1 + y2)/ 2

So here we have:

(-3 + 3)/ 2, (-4 + -1)/2

= 0/2, -5/2

= (0, -5/2).

The temperature at 9 AM is approximately 10 degrees. Neither of the answers are correct. (Correct choice: E)

Sinusoids are expressions with trascendent functions, to be more specfic, trigonometric functions, whose form is described below:

T(t) = A · sin (2π · t / T + C) + B       (1)

Where:

The temperature amplitude and the middle temperature can be found by the following expressions:

A = (t' - t'') / 2      (2)

B = (t' + t'') / 2      (3)

Where t' and t'' are maximum and minimum temperatures, in degrees.

Then, we proceed to find each constant of the sinusoidal model:

A = (45 - 7) / 2

A = 38 / 2

A = 19

B = (45 + 7) / 2

B = 52 / 2

B = 26

We assume a period of 24 hours (T = 24).

If we know that t = 0, T = 24, A = 19, B = 26, T(t) = 7, then the angular phase is:

7 = 19 · sin [(π / 12) · 0 + C] + 26

- 19 = 19 · sin C

sin C = - 1

C = π

T(t) = 19 · sin (π · t / 12 + π) + 26

Then, the temperature at 9 AM is: (t = 4)

T(4) = 19 · sin (π · 4 / 12 + π) + 26

T(4) ≈ 9.545

The temperature at 9 AM is approximately 10 degrees. Neither of the answers are correct. (Correct choice: E)

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Her profit % was increased by 2.77%

Answer:

Solution Given:

discount =10%

profit =25%

S.P with 13% vat = Rs 5,085

S.P+Vat% of S.P= Rs 5,085

S.P(1+13%)=Rs 5,085

S.P=Rs 5,085/1.13

S.P =Rs 4,500

Again

M.P = S.P+discount% of M.P

M.P- discount% of M.P= Rs 4,500

M.P(1-10%)=Rs 4,500

M.P=Rs 4,500/0.9

M.P = Rs 5,000

Now

C.P= (S.P*100)/(1+profit%)

C.P=(4,500*100)/(100+25%)

C.P= Rs 3,600

Again

profit =S.P- C.P= Rs 4,500-Rs 3,600=Rs 900

Again

Due to the excessive demands of her items, she decreased the discount percent by 2%.

So,

New discount = 10%-2%=8%

New S.P= M.P -discount% of M.P

=M.P(1-discount%)

=Rs 5,000(1-8%)

=Rs 4,600

Again

Profit=S.P- C.P

New profit =Rs 4,600 - Rs 3,600=Rs 1,000

Profit% =profit/C.P*100%= 1000/3600*100=27.77%

Profit % increased =New profit%- profit%

=27.77-25

=2.77%

Answer:

(-4, -18).

Step-by-step explanation:

f(x = x^2 + 8x - 2         Completing the square on x^2 + 8x:-

     = (x + 4)^2 - 16 - 2

     = (x + 4)^2 - 18

- which is the vertex form of f(x)

The vertex of

(x - h)^2 + k  is at (h, k)

So the vertex of f(x) is at (-4, -18)

Answer:

Step-by-step explanation:

a. d^2 - 7d + 6 = 0

(d - 1)(d - 6) = 0

d - 1 = 0,  d - 6 = 0 therefore:

d = 1,  6.

b, x^2 + 18x + 81 = 0

(x + 9)(x + 9) = 0

x = -9  multiplicity 2.

c   u^2 + 7u - 60 = 0

(u + 12)(u - 5) = 0

u = -12, 5.

Answer:

They are all true except for D

Step-by-step explanation:

What they are asking you to do, is take your rectangle and flip it over the top.  QR would stay right where it is and line ST would now be on top and line QR would be the bottom.  If you can picture that, you can see that all the statements would be true but D.

x = 3.5y equation shows the number of teaspoons of sugar, y, needed to make x tarts.

What is the linear equation in two variable?

Any equation which can be put in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero, is called linear equation in two variable.

According to the given question

A recipe require 3.5 teaspoons of sugar to make a tart.

y represents the number of teaspoons of sugar.

x represents the number of tarts.

Now, for x tart 3.5y teaspoons of sugar required.

⇒  x = 3.5y

Hence, the linear equation in two variable that shows the number of teaspoons of sugar y, needed to make x number or tarts is x = 3.5y.

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