naivlingr
2021-02-26
Answered

Solve the equation. Check your solution. $8t+5=6t+1$

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Nichole Watt

Answered 2021-02-27
Author has **100** answers

Subtract 6t from each side of the equation. Subtraction Property of Equality

$8t+5-6t=6t+1-6t$

Simplify

$2t+5=1$

Subtract 5 from each side of the equation. Subtraction Property of Equality

$2t+5-5=1-5$

Simplify

$2t=-4$

Divide each side of the equation by 2. Division Property of Equality

$\frac{2}{2}\ast t=\frac{-4}{2}$

Simplify

$t=-2$

Check solution

$8(-2)+5=6(-2)+1$

$-16+5=-12+1$

$-11=-11$

Simplify

Subtract 5 from each side of the equation. Subtraction Property of Equality

Simplify

Divide each side of the equation by 2. Division Property of Equality

Simplify

Check solution

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Let $C$ be the algebraic curve defined by the modular polynomial ${\varphi}_{N}$ of order $N>1$ over the rational numbers, i.e.

$C:=\text{specm}(\mathbb{Q}[X,Y]/{\varphi}_{N}(X,Y)).$

The singularities of this curve can be removed and we obtain a nonsingular curve ${C}^{sn}$n, then, we can embed ${C}^{sn}$ into a complete non-singular curve $\overline{C}$.

In Milne's notes "Modular Functions and Modular Forms" it is written:

The coordinate functions $x$ and $y$ are rational functions on $\overline{C}$, they generate the field of rational functions on $\overline{C}$, and they satisfy the relation ${\varphi}_{N}(x,y)=0$.

I assume, by coordinate functions he means the functions $f(X),g(X)\in \mathbb{Q}[X]$ such that ${\varphi}_{N}(f(x),g(x))=0$ for all $x\in Q$. However, I don't understand why the field of rational functions on $\overline{C}$ is generated by these functions. Could someone explain this to me?

Thank you very much in advance!

$C:=\text{specm}(\mathbb{Q}[X,Y]/{\varphi}_{N}(X,Y)).$

The singularities of this curve can be removed and we obtain a nonsingular curve ${C}^{sn}$n, then, we can embed ${C}^{sn}$ into a complete non-singular curve $\overline{C}$.

In Milne's notes "Modular Functions and Modular Forms" it is written:

The coordinate functions $x$ and $y$ are rational functions on $\overline{C}$, they generate the field of rational functions on $\overline{C}$, and they satisfy the relation ${\varphi}_{N}(x,y)=0$.

I assume, by coordinate functions he means the functions $f(X),g(X)\in \mathbb{Q}[X]$ such that ${\varphi}_{N}(f(x),g(x))=0$ for all $x\in Q$. However, I don't understand why the field of rational functions on $\overline{C}$ is generated by these functions. Could someone explain this to me?

Thank you very much in advance!

asked 2022-05-23

Prove that ${\mathrm{log}}_{9}15$ is irrational

Im having trouble with the following proof... Ill post what I have completed so far..

Ill attempt by contradiction assuming ${\mathrm{log}}_{9}15$ is rational.

So,

${\mathrm{log}}_{9}15=\frac{a}{b}$

$15={9}^{\frac{a}{b}}$

${15}^{b}={9}^{a}$ (This is where I'm getting stuck)

Any hints/tips/advice would be great. Thanks

Im having trouble with the following proof... Ill post what I have completed so far..

Ill attempt by contradiction assuming ${\mathrm{log}}_{9}15$ is rational.

So,

${\mathrm{log}}_{9}15=\frac{a}{b}$

$15={9}^{\frac{a}{b}}$

${15}^{b}={9}^{a}$ (This is where I'm getting stuck)

Any hints/tips/advice would be great. Thanks