Harvey is 3 times as old as Jane. The sum of their ages is 48 years. What is the age of each?

Jaylen Dudley
2022-09-12
Answered

Harvey is 3 times as old as Jane. The sum of their ages is 48 years. What is the age of each?

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asked 2021-06-01

Find the linear approximation of the function

Use L(x) to approximate the numbers

asked 2022-02-23

I had one doubt in matrix form of linear equations

Say , we have a system of equations$Ax=b$ such that A is an $n\times n$ matrix and b is a $n\times 1$ matrix and so is x then, if we are told that $\text{rank}\left(A\right)=n$ , then, do we need to check that $\text{rank}\left(A\mid b\right)=n$ or can we say that since b is a linear combination of the component vectors in A then augmenting it in A won't increase the number of linearly independent column vectors and hence the $\text{rank}\left(A\mid b\right)=n$ and can't ever be $n+1$ .

Say , we have a system of equations

asked 2022-05-19

While studying linear algebra I encountered the following result regarding the solutions of a non homogeneous system of linear equations: if $Sol(A,b\phantom{\rule{1px}{0ex}}b)\ne \mathrm{\varnothing}$

$Sol(A,b\phantom{\rule{1px}{0ex}}b)=x\phantom{\rule{1px}{0ex}}x+Sol(A,0\phantom{\rule{1px}{0ex}}0)$

Where x is a particular solution.

Then, while studying differential equations, I found that the solutions for

${y}^{\prime}=a(x)y+b(x)$

are all of the form

$y={y}_{p}+C{e}^{A(x)}$

Where the last term refers to the solutions of the associate homogeneous equation and ${y}_{p}$ is a particular solution.

These two concepts seem related to me, but I've not been able to find/think about a satisfying reason, so I'd like to know whether there is actually a relation or not.

I should specify that I've not been able to study properly systems of differential equations of the first order or equations of the n-th order yet, I suspect that the answer is there but I can't be sure. However I'd like to get the complete answer, so if it involves the latter topics it's not a problem.

$Sol(A,b\phantom{\rule{1px}{0ex}}b)=x\phantom{\rule{1px}{0ex}}x+Sol(A,0\phantom{\rule{1px}{0ex}}0)$

Where x is a particular solution.

Then, while studying differential equations, I found that the solutions for

${y}^{\prime}=a(x)y+b(x)$

are all of the form

$y={y}_{p}+C{e}^{A(x)}$

Where the last term refers to the solutions of the associate homogeneous equation and ${y}_{p}$ is a particular solution.

These two concepts seem related to me, but I've not been able to find/think about a satisfying reason, so I'd like to know whether there is actually a relation or not.

I should specify that I've not been able to study properly systems of differential equations of the first order or equations of the n-th order yet, I suspect that the answer is there but I can't be sure. However I'd like to get the complete answer, so if it involves the latter topics it's not a problem.

asked 2022-02-23

My calculus book says the equation:

$y}^{\prime}+{x}^{2}y={y}^{2$

is not linear.

Linear equations must be of the form:

${y}^{\prime}+P\left(x\right)y=Q\left(x\right)$

Is the equation not in this form with Q(x) = 0?

is not linear.

Linear equations must be of the form:

Is the equation not in this form with Q(x) = 0?

asked 2022-05-30

I am having trouble understanding the relatioship between rows and columns of a matrix.

Say, the following homogeneous system has a nontrivial solution.

$3{x}_{1}+5{x}_{2}-4{x}_{3}=0\phantom{\rule{0ex}{0ex}}-3{x}_{1}-2{x}_{2}+4{x}_{3}=0\phantom{\rule{0ex}{0ex}}6{x}_{1}+{x}_{2}-8{x}_{3}=0\phantom{\rule{0ex}{0ex}}$

Let A be the coefficient matrix and row reduce [A0] to row-echelon form:

$\left[\begin{array}{cccc}3& 5& -4& 0\\ -3& -2& 4& 0\\ 6& 1& -8& 0\end{array}\right]\to \left[\begin{array}{cccc}3& 5& -4& 0\\ 0& 3& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

$\phantom{\rule{1em}{0ex}}a1\phantom{\rule{1em}{0ex}}a2\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}a3$

Here, we see $\phantom{\rule{thinmathspace}{0ex}}{x}_{3}$ is a free variable and thus we can say 3rd column,$\phantom{\rule{thinmathspace}{0ex}}{a}_{3}$, is in $\text{span}({a}_{1},{a}_{2})$

But what does it mean for an echelon form of a matrix to have a row of 0's?

Does that mean 3rd row can be generated by 1st & 2nd rows?

just like 3rd column can be generated by 1st & 2nd columns?

And this raises another question for me, why do we mostly focus on columns of a matrix?

because I get the impression that ,for vectors and other concepts, our only concern is

whether the columns span ${\mathbb{R}}^{n}$ or the columns are linearly independent and so on.

I thought linear algebra is all about solving a system of linear equations,

and linear equations are rows of a matrix, thus i think it'd be logical to focus more on rows than columns. But why?

Say, the following homogeneous system has a nontrivial solution.

$3{x}_{1}+5{x}_{2}-4{x}_{3}=0\phantom{\rule{0ex}{0ex}}-3{x}_{1}-2{x}_{2}+4{x}_{3}=0\phantom{\rule{0ex}{0ex}}6{x}_{1}+{x}_{2}-8{x}_{3}=0\phantom{\rule{0ex}{0ex}}$

Let A be the coefficient matrix and row reduce [A0] to row-echelon form:

$\left[\begin{array}{cccc}3& 5& -4& 0\\ -3& -2& 4& 0\\ 6& 1& -8& 0\end{array}\right]\to \left[\begin{array}{cccc}3& 5& -4& 0\\ 0& 3& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

$\phantom{\rule{1em}{0ex}}a1\phantom{\rule{1em}{0ex}}a2\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}a3$

Here, we see $\phantom{\rule{thinmathspace}{0ex}}{x}_{3}$ is a free variable and thus we can say 3rd column,$\phantom{\rule{thinmathspace}{0ex}}{a}_{3}$, is in $\text{span}({a}_{1},{a}_{2})$

But what does it mean for an echelon form of a matrix to have a row of 0's?

Does that mean 3rd row can be generated by 1st & 2nd rows?

just like 3rd column can be generated by 1st & 2nd columns?

And this raises another question for me, why do we mostly focus on columns of a matrix?

because I get the impression that ,for vectors and other concepts, our only concern is

whether the columns span ${\mathbb{R}}^{n}$ or the columns are linearly independent and so on.

I thought linear algebra is all about solving a system of linear equations,

and linear equations are rows of a matrix, thus i think it'd be logical to focus more on rows than columns. But why?

asked 2022-09-08

Write linear equations for all lines that pass through point (0, 4)

asked 2022-09-07

A tennis court rental from 9am to 5pm is $3.50 per hour. After 5pm, the rate is $7.00 per hour. Shenice and her friends reserve a tennis court from 3pm to 8pm. What is the total court fees for this period?