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jekas [21]
3 years ago
5

What is the solution to 2x+y=1 and x-2y=3

Mathematics
1 answer:
Masteriza [31]3 years ago
5 0

Answer:

Step-by-step explanation:

2x+y=1---------> y=-2x+1

x-2y=3

2) x-2(-2x+1)=3                          

x+4x-2=3

5x=5

x=1

3) 2x+y=1 when x=1

2(1)+y=1

2+y=1

y=1

the answer is (1,1)

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(i) Yes. Simplify f(x,y).

\displaystyle \frac{x^2 - x^2y^2 + y^2}{x^2 + y^2} = 1 - \frac{x^2y^2}{x^2 + y^2}

Now compute the limit by converting to polar coordinates.

\displaystyle \lim_{(x,y)\to(0,0)} \frac{x^2y^2}{x^2+y^2} = \lim_{r\to0} \frac{r^4 \cos^2(\theta) \sin^2(\theta)}{r^2} = 0

This tells us

\displaystyle \lim_{(x,y)\to(0,0)} f(x,y) = 1

so we can define f(0,0)=1 to make the function continuous at the origin.

Alternatively, we have

\dfrac{x^2y^2}{x^2+y^2} \le \dfrac{x^4 + 2x^2y^2 + y^4}{x^2 + y^2} = \dfrac{(x^2+y^2)^2}{x^2+y^2} = x^2 + y^2

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\dfrac{x^2y^2}{x^2+y^2} \ge 0 \ge -x^2 - y^2

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\displaystyle \lim_{(x,y)\to(0,0)} -(x^2+y^2) = 0

\displaystyle \lim_{(x,y)\to(0,0)} (x^2+y^2) = 0

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and f(x,y) approaches 1 as we approach the origin.

(ii) No. Expand the fraction.

\displaystyle \frac{x^2 + y^3}{xy} = \frac xy + \frac{y^2}x

f(0,y) and f(x,0) are undefined, so there is no way to make f(x,y) continuous at (0, 0).

(iii) No. Similarly,

\dfrac{x^2 + y}y = \dfrac{x^2}y + 1

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Step-by-step explanation:

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sub and find b, the y int
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