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Elan Coil [88]
2 years ago
10

-4x-4y=24, 3x+10=y

Mathematics
1 answer:
Korvikt [17]2 years ago
3 0

-4x-4y = 24~~...(i)\\\\3x +10 = y~~....(ii)\\\\\text{Substitute}~ y = 3x +10~ \text{in eq (i):}\\\\-4x -4(3x+10) = 24\\\\\implies -4x -12x -40=24\\\\\implies -16x =24+40\\\\\implies -16x = 64\\\\\implies x =- \dfrac{64}{16}\\\\\implies x = -4\\\\\text{Substitute}~ x = -4 ~ \text{in equation (ii) to find y:} \\\\y=3(-4) +10 = -12 +10 = -2\\\\\text{Hence,}~(x,y) = (-4,-2)

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find the equation of a circle which passes through the point (2,-2) and (3,4) and whose centre lies on the line x+y=2
Nadusha1986 [10]

Answer:

Equation of the circle

(x - 0.7)² + (y - 1.3)² = 12.58

Step-by-step explanation:

The formula for the equation of a circle is given as:

(x - a)² + (y - b)² = r²,

where(a, b) is the center of the circle and r = radius of the circle.

a) We are told in the question that the equation of the circle passes through point(2, -2)

Hence,

Substituting 2 for x and -2 for y in the equation of the circle.

(x - a)² + (y - b)² = r²

(2 - a)² +(-2 - b)² = r²

Expanding the bracket

(2 - a) (2 - a) + (-2 - b)(-2 - b) = r²

4 - 2a - 2a +a² +4 +2b +2b +b² = r²

4 - 4a + a² + 4 + 4b + b² = r²

a² + b² -4a + 4b + 4 + 4 = r²

a² + b² -4a + 4b + 8 = r²............Equation 1

We are also told that the equation of the circle also passes through point (3,4) also, where 3 = x and 4 = y

Hence,

Substituting 3 for x and 4 for y in the equation of the circle.

(x - a)² + (y - b)² = r²

(3 - a)² +(4 - b)² = r²

Expanding the bracket

(3 - a) (3 - a) + (4 - b)(4- b) = r²

9 - 3a - 3a +a² +16 -4b -4b +b² = r²

9 -6a + a² + 16 -8b + b² = r²

a² + b² -6a -8b + 9 + 16 = r²

a² + b² -6a -8b + 25 = r²..........Equation 2

The next step would be to subtract Equation 1 from Equation 2

a² + b² -4a + 4b + 8 - (a² + b² -6a -8b + 25) = r² - r²

a² + b² -4a + 4b + 8 - a² - b² +6a +8b - -25= r² - r²

Collecting like terms

a² - a² + b² - b² - 4a + 6a + 4b + 8b +8- 25 = 0

2a + 12b -17 = 0

2a + 12b = 17...........Equation 3

Step 2

We are going to have to find the values of a and b in other to get our equation of the circle.

Since the center of the circle(a, b) lies on x + y = 2

Therefore, we have

a + b = 2

a = 2 - b

2a + 12b = 17 ..........Equation 3

Substituting 2 - b for a in

2(2 - b) + 12b = 17

4 - 2b + 12b = 17

4 + 10b = 17

10b = 17 - 4

10b = 13

b = 13/10

b = 1.3

Substituting 1.3 for b in

a + b = 2

a + 1.3 = 2

a = 2 - 1.3

a = 0.7

hence, a = 0.7, b = 1.3

Step 3

We have to find the value of r using points (2, -2)

(x - a)² + (y - b)² = r²

Where x = 2 and y = -2

(-2 - 0.7)² + (-2 - 1.3)² = r²

(-2.7)² + (-3.3)² = r²

1.69 + 10.89 = r²

r² = 12.58

r = √12.58 = 3.55

Step 4

The formula for the equation of a circle is given as:

(x - a)² + (y - b)² = r²,

where(a, b) is the center of the circle and r = radius of the circle

a = 0.7

b = 1.3

r² = 12.58

Equation of the circle =

(x - 0.7)² + (y - 1.3)² = 12.58

7 0
3 years ago
A culture started with 3000 bacteria. After 4 hours, it grew to 3,600 bacteria. Predict how many bacteria will be present after
Vlad [161]
Hello Shannonrodrigue. You can solve this by setting up an exponential growth equation.

A = A_ob^t
3600 = 3000b^4

Now we solve for b
3600 = 3000b^4
\frac{3600}{3000} = \frac{3000b^{4}}{3000}
\frac{6}{5} = b^4
\sqrt[4]{\frac{6}{5}} = \sqrt[4]{b^4}
\sqrt[4]{\frac{6}{5}} = b

Now that we have found b, we can use the equation A = 3000b^t to predict how many bacteria will be present after 10 hours.
b = \sqrt[4]{\frac{6}{5}}
t = 10
A = 3000b^t
A = 3000\sqrt[4]{\frac{6}{5}}^{10} = 4732.3228 = 4732

Answer = 4732



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\huge\boxed{12x-14}

Start with the original expression.

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Use the Distributive Property — multiply 2 by both 6x and -1.

-17+12x-2+5

Combine the like terms — add up all the constants.

-14+12x

Reorder the terms.

\large\boxed{12x-14}

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