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nydimaria [60]
2 years ago
12

Solve the simultaneous equations4x+3y=14x^2+3xy+y^2=2​

Mathematics
1 answer:
Fantom [35]2 years ago
8 0

Notice that the second equation,

4<em>x</em> ² + 3<em>xy</em> + <em>y</em> ² = 2

can be written as

<em>x</em> (4<em>x</em> + 3<em>y</em>) + <em>y</em> ² = 2

and the first equation says 4<em>x</em> + 3<em>y</em> = 1, so this reduces to

<em>x</em> + <em>y</em> ² = 2

Solve the first equation for <em>x</em> :

4<em>x</em> + 3<em>y</em> = 1

4<em>x</em> = 1 - 3<em>y</em>

<em>x</em> = (1 - 3<em>y</em>)/4

Substitute this into the reduced second equation to get a quadratic equation in <em>y</em>, which happens to be easily factorized and solved:

(1 - 3<em>y</em>)/4 + <em>y</em> ² = 2

1 - 3<em>y</em> + 4<em>y</em> ² = 8

4<em>y</em> ² - 3<em>y</em> - 7 = 0

(4<em>y</em> - 7) (<em>y</em> + 1) = 0

4<em>y</em> - 7 = 0   <u>or</u>   <em>y</em> + 1 = 0

<em>y</em> = 7/4   <u>or</u>   <em>y</em> = -1

Solve for <em>x</em> :

<em>x</em> = (1 - 3 (7/4))/4   <u>or</u>   <em>x</em> = (1 - 3 (-1))/4

<em>x</em> = -17/16   <u>or</u>   <em>x</em> = 1

So the two solutions are (<em>x</em>, <em>y</em>) = (-17/16, 7/4) and (<em>x</em>, <em>y</em>) = (1, -1).

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