Sasha Hess
2022-09-14
Answered

Find the slope of the line perpendicular to $y=-\frac{5}{12}x-5$

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Sharon Dawson

Answered 2022-09-15
Author has **20** answers

$y=-\frac{5}{12}x-5$

Compare y=mx+c

$\Rightarrow m=-\frac{5}{12}$

Slope of the given line is $-\frac{5}{12}$.

Let m' be the slope of line perpendicular to the given line.

If two lines are perpendicular then the product of their slopes is −1.

$\Rightarrow mm\prime =-1$

$\Rightarrow m\prime =-\frac{1}{m}=-\frac{1}{-\frac{5}{12}}=\frac{12}{5}$

$\Rightarrow m\prime =\frac{12}{5}$

Therefore, the slope of the line perpendicular to the given line is $\frac{12}{5}$.

Compare y=mx+c

$\Rightarrow m=-\frac{5}{12}$

Slope of the given line is $-\frac{5}{12}$.

Let m' be the slope of line perpendicular to the given line.

If two lines are perpendicular then the product of their slopes is −1.

$\Rightarrow mm\prime =-1$

$\Rightarrow m\prime =-\frac{1}{m}=-\frac{1}{-\frac{5}{12}}=\frac{12}{5}$

$\Rightarrow m\prime =\frac{12}{5}$

Therefore, the slope of the line perpendicular to the given line is $\frac{12}{5}$.

asked 2021-06-01

Find the linear approximation of the function

Use L(x) to approximate the numbers

asked 2022-02-25

The problem is:

The points (2,-1,-2), (1,3,12), and (4,2,3) lie on a unique plane. Where does this plane cross the z-axis?

I can easily solve this problem by calculus and cross product: the equation of a plane is

$-22x+33x-11z=-55.$

Hence, this plane crosses the z-axis at z=5. However, the problem requires to be sold by the system of linear equations transformed into matrix. Even vectors have not been introduces yet. Just a matrix in echelon form and a back substitution. Hence, when I am writing three equations:

$2a-b-2c=d$

$a+3b+12c=d$

$4a+2b+3c=d$

I end up with 4 variables to find. What am I missing? Is there one more linear equation possible to add?

The points (2,-1,-2), (1,3,12), and (4,2,3) lie on a unique plane. Where does this plane cross the z-axis?

I can easily solve this problem by calculus and cross product: the equation of a plane is

Hence, this plane crosses the z-axis at z=5. However, the problem requires to be sold by the system of linear equations transformed into matrix. Even vectors have not been introduces yet. Just a matrix in echelon form and a back substitution. Hence, when I am writing three equations:

I end up with 4 variables to find. What am I missing? Is there one more linear equation possible to add?

asked 2022-07-10

we have a system of linear equations as such:

$x+2y+(a-1)z=1\phantom{\rule{0ex}{0ex}}-x-y+z=0\phantom{\rule{0ex}{0ex}}-ax-(a+3)y-az=-3\phantom{\rule{0ex}{0ex}}-ax-(a+2)y+0\cdot z={a}^{2}-5a-2$

and i have to find the solution in $\mathbb{R}$ and ${\mathbb{Z}}_{\mathbb{5}}$ so i have no problem for $\mathbb{R}$ i get the matrix

$\left(\begin{array}{cccc}1& 2& a-1& 1\\ 0& 1& a& 1\\ 0& 0& a& 0\\ 0& 0& a& {a}^{2}-5\cdot a\end{array}\right)$

but the questions i have are as follows:

1. Can i use what i found for the augmented matrix and the discussion by parameter a in $\mathbb{R}$ to deduce ${\mathbb{Z}}_{\mathbb{5}}$?

2. Or is there some other way i must reduce to row echelon form for ${\mathbb{Z}}_{\mathbb{5}}$ and then have the discussion for parameter a?

3. If i had an 3x3 or 4x4 system to solve over a low prime ${\mathbb{Z}}_{{\mathbb{p}}_{\mathbb{1}}}$ and ${\mathbb{Z}}_{{\mathbb{p}}_{\mathbb{2}}}$ (eg 5 and 7) how would i go about doing it with the matrix gauss elimination?could i use the same augmented matrix and reduce it to row echelon over $\mathbb{R}$ and then use that augmented matrix for the rest like above or not?

4. If i recall correctly there was a theorem about the rank of the original matrix and augmented that says something about the number of solutions but i do not recall how that would help me find solutions just eliminate the a's where there is none?

$x+2y+(a-1)z=1\phantom{\rule{0ex}{0ex}}-x-y+z=0\phantom{\rule{0ex}{0ex}}-ax-(a+3)y-az=-3\phantom{\rule{0ex}{0ex}}-ax-(a+2)y+0\cdot z={a}^{2}-5a-2$

and i have to find the solution in $\mathbb{R}$ and ${\mathbb{Z}}_{\mathbb{5}}$ so i have no problem for $\mathbb{R}$ i get the matrix

$\left(\begin{array}{cccc}1& 2& a-1& 1\\ 0& 1& a& 1\\ 0& 0& a& 0\\ 0& 0& a& {a}^{2}-5\cdot a\end{array}\right)$

but the questions i have are as follows:

1. Can i use what i found for the augmented matrix and the discussion by parameter a in $\mathbb{R}$ to deduce ${\mathbb{Z}}_{\mathbb{5}}$?

2. Or is there some other way i must reduce to row echelon form for ${\mathbb{Z}}_{\mathbb{5}}$ and then have the discussion for parameter a?

3. If i had an 3x3 or 4x4 system to solve over a low prime ${\mathbb{Z}}_{{\mathbb{p}}_{\mathbb{1}}}$ and ${\mathbb{Z}}_{{\mathbb{p}}_{\mathbb{2}}}$ (eg 5 and 7) how would i go about doing it with the matrix gauss elimination?could i use the same augmented matrix and reduce it to row echelon over $\mathbb{R}$ and then use that augmented matrix for the rest like above or not?

4. If i recall correctly there was a theorem about the rank of the original matrix and augmented that says something about the number of solutions but i do not recall how that would help me find solutions just eliminate the a's where there is none?

asked 2021-03-05

Find the augmented matrix for the following system of linear equations:

asked 2021-09-14

Determine whether each statement makes sense or does not make sense, and explain your reasoning. A system of linear equations in three variables, x, y, and z cannot contain an equation in the form y = mx + b.

asked 2022-02-24

Theorem:

Given a system of linear equations$Ax=b$ where

$A\in {M}_{m\times n}\left(\mathbb{R}\right),x\in {\mathbb{R}}_{\text{col}}^{n},b\in {\mathbb{R}}_{\text{col}}^{m}$ .

Deduce that a solution x exists if and only if$\text{rank}\left(A\mid b\right)=\text{rank}\left(A\right)$ where $A\mid b$ is the augmented coefficient matrix of this system

I am having trouble proving the above theorem from my Linear Algebra course, I understand that A|b must reduce under elementary row operations to a form which is consistent but I don't understand exactly why the matrix A|b need have the same rank as A for this to happen.

Please correct me if I am mistaken

Given a system of linear equations

Deduce that a solution x exists if and only if

I am having trouble proving the above theorem from my Linear Algebra course, I understand that A|b must reduce under elementary row operations to a form which is consistent but I don't understand exactly why the matrix A|b need have the same rank as A for this to happen.

Please correct me if I am mistaken

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.