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2 years ago

Use substitution to solve the following system of equations. - 2x + 4y = -18AND x = y + 3

Mathematics

1 answer:

Dima020 [189]2 years ago

7 0

The anwser is (-6,-9)

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A circle has the order pairs (-1, 2) (0, 1) (-2, -1) what is the equation . Show your work.

olga55 [171]

We know that:

$(x-a)^2+(y-b)^2=r^2$

is an equation of a circle.

When we substitute x and y (from the pairs we have), we'll get a system of equations:

$\begin{cases}(-1-a)^2+(2-b)^2=r^2\\(0-a)^2+(1-b)^2=r^2\\(-2-a)^2+(-1-b)^2=r^2\end{cases}$

and all we have to do is solve it for a, b and r.

There will be:

$\begin{cases}(-1-a)^2+(2-b)^2=r^2\\(0-a)^2+(1-b)^2=r^2\\(-2-a)^2+(-1-b)^2=r^2\end{cases}\\\\\\
\begin{cases}1+2a+a^2+4-4b+b^2=r^2\\a^2+1-2b+b^2=r^2\\4+4a+a^2+1+2b+b^2=r^2\end{cases}\\\\\\
\begin{cases}a^2+b^2+2a-4b+5=r^2\\a^2+b^2-2b+1=r^2\\a^2+b^2+4a+2b+5=r^2\end{cases}\\\\\\
$

From equations (II) and (III) we have:

$\begin{cases}a^2+b^2-2b+1=r^2\\a^2+b^2+4a+2b+5=r^2\end{cases}\\--------------(-)\\\\a^2+b^2-2b+1-a^2-b^2-4a-2b-5=r^2-r^2\\\\-4a-4b-4=0\qquad|:(-4)\\\\\boxed{-a-b-1=0}$

and from (I) and (II):

$\begin{cases}a^2+b^2+2a-4b+5=r^2\\a^2+b^2-2b+1=r^2\end{cases}\\--------------(-)\\\\a^2+b^2+2a-4b+5-a^2-b^2+2b-1=r^2-r^2\\\\2a-2b+4=0\qquad|:2\\\\\boxed{a-b+2=0}$

Now we can easly calculate a and b:

$\begin{cases}-a-b-1=0\\a-b+2=0\end{cases}\\--------(+)\\\\-a-b-1+a-b+2=0+0\\\\-2b+1=0\\\\-2b=-1\qquad|:(-2)\\\\\boxed{b=\frac{1}{2}}\\\\\\\\a-b+2=0\\\\\\a-\dfrac{1}{2}+2=0\\\\\\a+\dfrac{3}{2}=0\\\\\\\boxed{a=-\frac{3}{2}}$

Finally we calculate $r^2$ :

$a^2+b^2-2b+1=r^2\\\\\\\left(-\dfrac{3}{2}\right)^2+\left(\dfrac{1}{2}\right)^2-2\cdot\dfrac{1}{2}+1=r^2\\\\\\\dfrac{9}{4}+\dfrac{1}{4}-1+1=r^2\\\\\\\dfrac{10}{4}=r^2\\\\\\\boxed{r^2=\frac{5}{2}}$

And the equation of the circle is:

$(x-a)^2+(y-b)^2=r^2\\\\\\\left(x-\left(-\dfrac{3}{2}\right)\right)^2+\left(y-\dfrac{1}{2}\right)^2=\dfrac{5}{2}\\\\\\\boxed{\left(x+\dfrac{3}{2}\right)^2+\left(y-\dfrac{1}{2}\right)^2=\dfrac{5}{2}}$

7 0

3 years ago

Number 18. Please?!!!!

Ksju [112]

Answer:

The womans age is 25

mothers age is 50

Step-by-step explanation:

add those up and you get 75

8 0

1 year ago

Read 2 more answers

What’s 3х + 7>4х +2

Kipish [7]

Answer:

$x < 5$

Step-by-step explanation:

$3x + 7 > 4x + 2 \\ 7 - 2 > 4x - 3x \\ 5 > x$

$x < 5$

6 0

3 years ago

Help plz i’ll give extra points f

lys-0071 [83]

Answer: First Option | 2 x 10^1

Reason: After doing 6.4 divided by 3.2, you get 2, and finally you subtract 10³ by 10². And after that subtraction, you get 10^1, Therefore the answer is the first option.

<em>I hope this helps, and Happy Holidays! :)</em>

3 0

2 years ago

What is the percentage increase of a room that is the size of 96 square feet to a room that is 108 square feet?

katrin [286]

Answer:

12.5% increase OR 112.5% Percentage of Change

Step-by-step explanation:

For this problem consider that our original maximum value is 96 square feet, but our new maximum value is 108 square feet. So to find the change as a percentage (in this case the increase) use the following formula:

Percentage of Change = ( New Maximum / Old Maximum ) * 100

So, let's use this formula to find the percentage change of the room.

Percentage of Change = ( 108 / 96 ) * 100

Percentage of Change = (1.125) * 100

Percentage of Change = 112.5

So the percentage of Change is 112.5%. Note, the old maximum is the point of comparison which is 100%.

So to find the increase, we will do 112.5% - 100% to get 12.5%. Hence, we have a 12.5% increase of a 108 square foot room compared to a room of 96 square feet.

Cheers.

6 0

2 years ago