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Semmy [17]
2 years ago
11

Write the equation in standard form. 8−5y=−2x

Mathematics
1 answer:
MrMuchimi2 years ago
4 0

bearing in mind that

standard form for a linear equation means

• all coefficients must be integers, no fractions

• only the constant on the right-hand-side

• all variables on the left-hand-side, sorted

• "x" must not have a negative coefficient


8 - 5y = - 2x

2x - 5y = - 8

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Solve the following equations: (a) x^11=13 mod 35 (b) x^5=3 mod 64
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a.

x^{11}=13\pmod{35}\implies\begin{cases}x^{11}\equiv13\equiv3\pmod5\\x^{11}\equiv13\equiv6\pmod7\end{cases}

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x^{11}\equiv (x^5)^2x\equiv x^3\equiv3\pmod5

x^{11}\equiv x^7x^4\equiv x^5\equiv6\pmod 7

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x^5\equiv3\pmod{64}

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x^{32}\equiv1\pmod{64}

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x^{30}\equiv3^6\equiv729\equiv25\pmod{64}

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x^{32}\equiv25x^2\equiv1\pmod{64}

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14 = 1*11 + 3

11 = 3*3 + 2

3 = 1*2 + 1

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so that (-23)\cdot25\equiv1\pmod{64}\implies y=25^{-1}\equiv-23\equiv41\pmod{64}.

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25x^2\equiv1\pmod{64}\implies x^2\equiv41\pmod{64}

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x^4\equiv1681\equiv17\pmod{64}

and multiplying both sides by x tells us

x^5\equiv17x\equiv3\pmod{64}

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13 = 3*4 + 1

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