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SVETLANKA909090 [29]
3 years ago
8

Help with line equations?

Mathematics
1 answer:
Alecsey [184]3 years ago
5 0
First question

line:~~~~y=x+3

curve:~~~~x^2+y^2=29

now we have to replace the y of curve by the y of line, therefore

x^2+(x+3)^2=29

x^2+x^2+6x+9=29

2x^2+6x+9-29=0

2x^2+6x-20=0

we can multiply each member by \frac{1}{2}

\boxed{\boxed{x^2+3x-10=0}}

Now we have to find the roots of this funtion

x^2+3x-10=0

Sum and produc or Bhaskara

Then we find two axis

\boxed{x_1=2~~and~~x_2=-5}

now we replace this values to find the Y, we can replace in curve or line equation, I'll prefer to replace it in line equation.

y=x+3

y_1=x_1+3

y_1=2+3

\boxed{y_1=5}

y_2=x_2+3

y_2=-5+3

\boxed{y_2=-2}

Therefore

\boxed{\boxed{P_1(2,5)~~and~~P_2(-5,-2)}}
_______________________________________________________________

The second question give to us

y=ax+b

P_1(2,13)

P_2(-1,-11)

We just have to replace the value then we'll get a linear system.

point 1

13=2a+b

point 2

-11=-a+b

then our linear system will be

\begin{Bmatrix}2a+b&=&13\\-a+b&=&-11\end{matrix}

I'll multiply the second line by -1 and I'll add to first one

\begin{Bmatrix}2a+(a)+b+(-b)&=&13+(-11)\\-a+b&=&-11\end{matrix}

\begin{Bmatrix}3a&=&24\\-a+b&=&-11\end{matrix}

\begin{Bmatrix}a&=&8\\-a+b&=&-11\end{matrix}

therefore we can replace the value of a, at second line

\begin{Bmatrix}a&=&8\\-8+b&=&-11\end{matrix}

\begin{Bmatrix}a&=&8\\b&=&-11+8\end{matrix}

\boxed{\boxed{\begin{Bmatrix}a&=&8\\b&=&-3\end{matrix}}}

then our function will be

\boxed{\boxed{y=8x-3}}

_________________________________________________________________

The third one, we have

line:~~~~y=3x-4

curve:~~~~y=x^2-2x-4

This resolution will be the same of our first question.

Let's replace the y of curve by the y of line

3x-4=x^2-2x-4

0=x^2-2x-4-3x+4

therefore

\boxed{\boxed{x^2-5x=0}} 

now we have to find the roots of this function.

x^2-5x=0

put x in evidence

x*(x-5)=0

\boxed{x_1=0~~and~~x_2=5}

then

y=3x-4

y_1=3x_1-4

y_1=3*0-4

\boxed{y_1=-4}

y_2=3x_2-4

y_2=3*5-4

y_2=15-4

\boxed{y_2=11}

our points will be

\boxed{\boxed{P_1(0,-4)~~and~~P_2(5,11)}}

I hope you enjoy it ;)
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