Master Algebra 1 Word Problems with our Comprehensive Practice

Is linear constraint a constraint in form of linear equation?
I used equation ${\displaystyle f_i(\alpha x+\beta y)=\alpha f_i\left(x\right)+\beta f_i\left(y\right)}$ to test linearity of constraint. For ${\displaystyle f_i\left(x\right)=x+5}$ test failed. For ${\displaystyle f_i\left(x\right)=2x}$ test passed.
What I don't get is x+5 is linear equation then, why it failed the test.

${\displaystyle x(k+2)+y(k-1)+z\left(k\right)=2}$
${\displaystyle y(k+2)+2x=0}$
${\displaystyle z(k^2+k-2)=k+2}$
Is there any value of k for which this system of linear equations would have infinite solutions? I mean, it seems as if it does when k = -2, but it also seems as if it has no solutions when k = -2. Im

From a text book:
"The general form of a linear equation in two variables is ${\displaystyle ax+by+c=0}$ or, ${\displaystyle ax+by=c}$ where a,b,c are real numbers such that ${\displaystyle a\ne 0,b\ne 0}$ and x,y are variables. (we often denote the condition a and b are not both zero by ${\displaystyle a^2+b^2\ne 0}$.)"
I don’t understand this last condition.
How can we say that ${\displaystyle a^2+b^2\ne 0}$ represents the condition that a and b are not both zero.
Let a=0,b=1, then also this condition fulfills.

Write the prime factorization of 280 using exponents.

I am looking for general solution of the form ${\displaystyle B=XA{X}^{-1}}$ with the following constraints
${\displaystyle X^{-1}=X^{\prime }}$
where X is unknown matrix and A,B,X, are ${\displaystyle 3\times 3}$ matrices.
The problem appears peculiar and thus I wish to simplify it further to more practical problem X is rotation matrix (R), and A is plane equation matrix, B is resulted homography in the sense of
${\displaystyle B=XA{X}^{-1}=XA{X}^T=R(I-\frac{t{n}^T}{d})R^T}$,
where t is translation vector, n is plane normal vector, and d is signed distance, all three are known
My dirty solution so far is as follows : Finding the Jordan Canonical form:
${\displaystyle MB{M}^{-1}=J=NA{N}^{-1}}$
and then X is obviously
${\displaystyle X=N^{-1}M}$
but this works only if the canonical forms are the same, otherwise cannot be utilized for solution.
The second dirty trick is to solve a system of Linear equations in terms of
${\displaystyle AX=BX}$
but i cannot figure out how to program this in the software (e.g. Matlab linsolve). Could you help me to clarify how to write system of linear equations in the shape of 9 rows?
The trick is how to notate them on the "right side" to use in ready to use functions for solving SLE.

Let ${\displaystyle {\lambda }_1,{\lambda }_2,{\lambda }_3,b\in \mathbb{R}}$ with ${\displaystyle {\lambda }_1\ne 0}$. Show that the linear equation ${\displaystyle {\lambda }_1{x}_1+{\lambda }_2{x}_2+{\lambda }_3{x}_3=b}$ has a set of solutions of the form ${\displaystyle \left\{v\right\}+\text{span}(v_1,v_2)}$, with ${\displaystyle v,v_1,v_2\in {\mathbb{R}}^3}$ and ${\displaystyle v_1,v_2}$ are linearly independent.
Could you give me a hint how we could show that, since I have no idea? I don't really know how to start.

How to solve the system of linear differential equation of the form
${\displaystyle x^{\prime }=Ax+b}$
I can solve the homogeneous form by finding the eigenvalues and respective eigenvectors, but how to find the particular solution part. Also is there any limitation from getting eigenvalues positive, negative or complex. Any other different method involving matrix algebra is also welcome.

We know the standard form of expressing a system of linear equations in n variables in n equations.
${\displaystyle A_{n\times n}\cdot X_{n\times 1}=B_{n\times 1}}$
Where A is the coefficient matrix, X is the unknown variables (each variable maybe a real or complex number) matrix and B is the constants matrix.
Now, arriving at the question, let's say I have a set of n vectors X, each of length n and another set of n+1 vectors Y, each of length n. (Clearly, ${\displaystyle size\left(X\right)<size\left(Y\right)}$). All elements of X are independent. The same goes for Y.
Now I want to express each element in Y, as a linear expression of all the elements in X. So that would be something like below, for some coefficient matrix K.
${\displaystyle K_{(n+1)\times n}\cdot X_{n\times n}=Y_{(n+1)\times n}}$
My question is, is it necessary that K must always exist i.e. there must be a way to express all elements in Y as a linear combination of X.
I think the answer is No. The reasoning is that, for a system of linear equations where number of equations is strictly greater than the number of unknowns, there does not exist a solution at all. In the above case, we have n unknowns from X and n+1 equations, each one for each element in Y.
Is my answer and reasoning correct?

How can we derive the standard form of the linear equation:
${\displaystyle Ax+By+C=0}$? What do "A", "B" and "C" in the standard form of the linear equation mean? As in the point-slope form of the linear equation:
${\displaystyle y-y_0=m(x-x_0)}$, m is the slope, and x, y are the coordinates of any point. Likewise,${\displaystyle x_0,y_0}$, are the coordinates of the given point.

I have a set of linear equations, of the form Ax=b . However, the value of b depends on the sign of the solution of x, such that
${\displaystyle b_i=c_i+d_i}$ if ${\displaystyle x_i>0}$, and ${\displaystyle b_i=c_i-d_i}$ if ${\displaystyle x_i<0}$.
I could solve this by trying each combination of positiveness of x, but that would require solving ${\displaystyle 2^N}$ sets of linear equations, which becomes unfeasible quite quickly.
Is this solvable analytically?

I was stuck at numbers 2 and 6. This is a linear ODE.
${\displaystyle 2.(y+1)dx+(4x-y)dy=0}$
Divided it by ${\displaystyle dx}$:
${\displaystyle \frac{(y+1)+(4x-y)dy}{dx}=0}$
Transposed the ${\displaystyle (y+1)}$
${\displaystyle \frac{(4x-y)dy}{dx}=-(y+1)}$
Divided it by ${\displaystyle (4x-y)}$
${\displaystyle \frac{dy}{dx}=2y(x-3)}$
The farthest I can go is:
${\displaystyle \frac{dx}{dy}=\left(\frac{2x}{3x^2}\right)-\left(\frac{6y}{3x^3}\right)}$.
The rest of the problems were easily solved but those 2 are very hard to bring to the standard form.

If you have a system fo linear equations, the solution is where the equations intersect
Using row reduction you get a system of linear equations which still satisfies the intersection
Why are free variables used?
The values in which the free variable can be are limited in a range as for if the solution is a line and not a plane
So what is the point in writing the solutions as a set of vector additions using free variables?
If z in this case is a free variable, writing the solution in a vector equation
Where , denotes a new row
Solving for the pivot variables x and y and writing into decomposed vector form
[x,y] = [4,0] - [0,3] + z[1,2]
Z is said to be any real number zeR But looking at the graph, when z = 4, y=11/8 is not on the line of intersection So why do they say that z is a free variable when it isnt?

I have a very simple question: suppose I have two 2D linear equations in general form
${\displaystyle a_{1}x+b_{1}y+c_1=0}$
${\displaystyle a_{2}x+b_{2}y+c_2=0}$
Id

General form of linear equation using two variables I always read can be written:
${\displaystyle Ax+By+c=0}$
where A and B cannot both be 0.
So does it means like ${\displaystyle 1x+0y+c=0}$
is a linear equation of two variables i.e basically what I am asking is that both A and B cannot be zero but can either A or B be 0 like:
${\displaystyle 0x+4y-5=0}$ or ${\displaystyle 3x+0y-15=0}$
Are the above equation a Linear equation of 2 variables?
Need your help!

I have some question to ask you that explain to me and remove my confusion.
1.The book said that we call for those equations linear that have first degree power or general forms of linear equations now my question is that we call for those kind of equations linear that have general form of linear equation or can change to general form or other forms of linear equations form example can we call ${\displaystyle \frac{10}{x}=5}$ linear or square root of ${\displaystyle (y-9)=10}$ linear?
2.Can those rational and radical equations that can be manipulated to linear forms called linear equations like the examples above?
3.Are all first degree one variable equations linear?

Common form of system of linear equations is A*X = B, X is unknown. But how to find A, if X and B are known?
A is MxN matrix, X is column vector(N), B is column vector(M)

The linear equation is the equation that has a from:
${\displaystyle a_1{x}_1+a_2{x}_2+\dots +a_n{x}_n=b}$
But is this really the definition of a linear equation? I thought that the definition should be the form of linear mapping. Like,
${\displaystyle f(x+y)=f\left(x\right)+f\left(y\right)}$
${\displaystyle f\left(ax\right)=af\left(x\right)}$
What is the definition of a linear equation?

My book says the general form of a linear differential equation is:
${\displaystyle y^{\prime }+P\left(x\right)y=Q\left(x\right)}$
But my teacher said that a linear differential equation has the general form:
${\displaystyle y{}^{″}+P\left(x\right)y^{\prime }+Q\left(x\right)y=f\left(x\right)}$
Which is the correct form or they are both correct?

Consider the sunction f(x)-x^1/2 and the function g, shown below

Perform the multiplication or division and simplify.
${\displaystyle \frac{x+3}{4x^2-9}÷\frac{x^2+7x+12}{2^2+7x-15}}$