treallt5
2022-09-14
Answered

Find the slope perpendicular to 5x+2y=10

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asked 2021-06-01

Find the linear approximation of the function

Use L(x) to approximate the numbers

asked 2022-06-25

So my course shows me three differential equations:

$\dot{x}+{x}^{2}=t$

$\dot{x}=({t}^{2}+1)(x-1)$

$\dot{x}+x={t}^{2}$

The first one is not a linear ordinary differential equation (ODE) apparently, the other two are.

Unfortunately, they don't show a clear way how to find out if an ODE is linear or not. So how we can find out if an ODE is linear?

For the second one, I thought I bring it into standard form somehow:

$\dot{x}=({t}^{2}+1)(x-1)=x{t}^{2}+x-{t}^{2}-1=x({t}^{2}+1)-{t}^{2}-1$

If we say we let $p(t)={t}^{2}+1$ and $q(t)=1+{t}^{2}$, then we could say:

$\dot{x}=xp(t)-q(t)=...$

And so on, to simplify until we reach standard form of a linear ODE (or not).

Is that the way to go? Or is there some other way to check if a ODE is linear?

$\dot{x}+{x}^{2}=t$

$\dot{x}=({t}^{2}+1)(x-1)$

$\dot{x}+x={t}^{2}$

The first one is not a linear ordinary differential equation (ODE) apparently, the other two are.

Unfortunately, they don't show a clear way how to find out if an ODE is linear or not. So how we can find out if an ODE is linear?

For the second one, I thought I bring it into standard form somehow:

$\dot{x}=({t}^{2}+1)(x-1)=x{t}^{2}+x-{t}^{2}-1=x({t}^{2}+1)-{t}^{2}-1$

If we say we let $p(t)={t}^{2}+1$ and $q(t)=1+{t}^{2}$, then we could say:

$\dot{x}=xp(t)-q(t)=...$

And so on, to simplify until we reach standard form of a linear ODE (or not).

Is that the way to go? Or is there some other way to check if a ODE is linear?

asked 2021-05-31

Do the equations

asked 2022-09-05

Again with this!! I need a sum of 2 numbers that equals 20 and they both have a difference of 48, can you please help me? Also, i can't seem to understand this so could you kinda help with that also?

asked 2022-04-10

Emma is planning her summer and would like to work enough to travel and buy a new laptop. She can earn 90 dollars each day, after deductions, and she can work a maximum of 40 days in July and August, combined. She expects each day of travel will cost her 150 dollars and the laptop she hopes to buy costs 700 dollars.

Write a linear equation that represents the number of days Emma can work and travel and still earn enough for her laptop.

This is what I've come up with: 90d = 150p + 700, (Standard form: 90d - 150p -700 = 0) d represents days worked, and p represents travel days.

The question goes on to ask about how many days she will need to work she wants to travel, so I want to make sure i have to correct equation before I answer.

Write a linear equation that represents the number of days Emma can work and travel and still earn enough for her laptop.

This is what I've come up with: 90d = 150p + 700, (Standard form: 90d - 150p -700 = 0) d represents days worked, and p represents travel days.

The question goes on to ask about how many days she will need to work she wants to travel, so I want to make sure i have to correct equation before I answer.

asked 2022-02-25

Let $A\in {M}_{m\times n}\left(\mathbb{Q}\right)$ and $b\in {\mathbb{Q}}^{m}$ . Suppose that the system of linear equations Ax=b has a solution in $\mathbb{R}}^{n$ . Does it necessarily have a solution in $\mathbb{Q}}^{n$ ?

and I thought I'd give an interesting, possibly wrong, approach to solving it. I'm not sure if such things can be done, if not maybe you can help me refine.

I considered the form of the equality as

${A}^{\left(1\right)}{x}_{1}+\cdots +{A}^{\left(n\right)}{x}_{n}=b$

where$A}^{\left(i\right)$ is a column vector of A. I then noticed that for $x}_{i}\in \mathbb{R}\mathrm{\setminus}\mathbb{Q}=\mathbb{T$ then, and this is where I think I'm doing something forbidden, each x has the represenation

$x}_{1}={k}_{11}{\tau}_{1}+\cdots +{k}_{1p}{\tau}_{p$

$x}_{2}={k}_{21}{\tau}_{1}+\cdots +{k}_{2p}{\tau}_{p$

$\vdots$

$x}_{n}={k}_{n1}{\tau}_{1}+\cdots +{k}_{np}{\tau}_{p$ ,

where$\tau}_{i$ is a distinct irrational number, $k}_{ij}\in \mathbb{R$ , and p is the number of such distinct irrational numbers. I wound this out, but there may be a discrepancy with p and m. I feel this method can lead me to the answer, but I'm not sure where to go from here.

I end up getting something like this, I believe, after substitution:

$A({k}^{\left(1\right)}{\tau}_{1}+\cdots +{k}^{\left(p\right)}{\tau}_{p})=b$

Here,$k}^{\left(i\right)$ is the vector

$(({k}_{1i}),(\dots ),({k}_{ni}))$ .

I think there is no discrepancy with p and m because$A\in {M}_{m\times n}\left(\mathbb{Q}\right),K\in {M}_{n\times p}\left(\mathbb{R}\right)$ , and $\tau \in {M}_{p\times 1}\left(\mathbb{T}\right)$ , so

$(m\times n)\cdot (n\times p)\cdot (p\times 1)=m\times 1$ .

and I thought I'd give an interesting, possibly wrong, approach to solving it. I'm not sure if such things can be done, if not maybe you can help me refine.

I considered the form of the equality as

where

where

I end up getting something like this, I believe, after substitution:

Here,

I think there is no discrepancy with p and m because

asked 2022-02-22

Question: Solve the system of linear congruences below by finding all x that satisfy it. Hint — try rewriting each congruence in the form $x\equiv a\left(\text{mod}\text{}b\right)$ .

$2x\equiv 1\left(\text{mod}\text{}3\right)$

$3x\equiv 2\left(\text{mod}\text{}5\right)$

$5x\equiv 4\left(\text{mod}\text{}7\right)$

So I tried following that hint, and I have:

$x\equiv 2\left(\text{mod}\text{}3\right)$

$x\equiv 4\left(\text{mod}\text{}5\right)$

$x\equiv 5\left(\text{mod}\text{}7\right)$

Firstly, is this correct? Second, where do I go from here? I calculated that$M=3\times 5\times 7=105$ and the individual $M}_{i$ 's, but now I'm stuck in a circle, because reducing from there gives me back the original equations. Any tips?

So I tried following that hint, and I have:

Firstly, is this correct? Second, where do I go from here? I calculated that