Master Algebra 1 Word Problems with our Comprehensive Practice

1000 tickets were sold. Adult tickets cost $8.50, children's tickets cost $4.50, and a total of $7300 was collected. How many tickets of each kind were sold?

Find a solution for a differential equation, assume that $y(x)=a{x}^2+bx+c$
Find a solution to the given equation:
$y^{\prime }+2y=x^2+4x+7$
and we are told to assume that y(x) is a quadratic function, which follows this general form
$y=a{x}^2+bx+c.$
Now to find the solutions, at first I tried to solve the given for y and then see what I could do from there but, I realized that you can't take the derivative of a function that contains its own derivative.
So then I thought that I had to create my own function that follows the perviously mentioned parameters (the given and the quad. form). This is were I am stuck, I can not mathematically deduce a solution for this problem.

Find the form of f(x, y) knowing the form of contour lines in the XZ and YZ planes
I am trying to find the form of $z=f(x,y)$. I know that:
Contour lines in the XZ plane are of the form:
$z=A\ast <!--\; \ast \; -->ln(x)+B$
(the A and B parameters vary with y)
Contour lines in the YZ plane are of the form:
$z=C{y}^2+Dy+E$
(the C, D and E parameters vary with x)
What is then the form of $z=f(x,y)$?

Location of Roots in Quadratic Equation
We learnt about the location of roots of quadratic equation in class today. I have a problem in one case specifically, the one where both the roots are required in the interval $(k_1,k_2)$ where $k_1,k_2$ are real constant numbers. So the conditions given to us were,
1. $\mathrm{\Delta <!--\; \Delta \; -->}$ greater than or equal to 0 (fairly obvious)
2. $af(k_1)>0$ where $f(k_1)$ is the value of the function at $k_1$ and a is the leading coefficient
3. $af(k_2)>0$ (both of these are again understandable)
4. then comes $k_1<-<!--\; -\; -->b/2a<k_2$ i.e. the abscissa of the vertex of the parabola lies in the interval required.

Lyapunov equation for stability analysis - what's the point?
In the following theorem $A,P,Q\in <!--\; \in \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->n}$, and P and Q are symmetric. The notation $P>0$ means that the matrix P is positive definite.
Given any $Q>0$, there exists a unique $P>0$ satisfying $A^{T}P+PA+Q=0$ if and only if the linear system $\stackrel{\dot{}<!--\; \dot{}\; -->}{x}=Ax$ is globally asymptotically stable. The quadratic function $V(z)=z^{T}Pz$ is a Lyapunov function that can be used to verify stability.
Recently I became quite familiar with tools to solve Lyapunov equation $A^{T}P+PA+Q=0$ and obtain P. However in the context of stability analysis I see one huge problem. It's that you still have to check that P is positive definite or equivalently that its eigenvalues are positive. But for the linear system the fact that all eigenvalues of A have negative real parts ensures that the system is globally asymptotically stable in the first place.
So what's the point of going through all the hassle just to end up with looking for the eigenvalues (P and A are of the same size!) anyway? Does the symmetric matrix make such a big difference?

${\displaystyle -6\le 2(x-5)<7}$

Tell whether the sequence 5,10,20,40,80,... is geometric

Let $X=\mathbb{R}$ and $S$ be the set of irrational numbers. Find the closure of $S$ in the following topologies for $R$
The topology of the accounting complement

Let $q$ be irrational. Let $a,b$ be rationals such that $a<q<b$. If there exists an element $p$ s.t. $a<p<b$, would it be valid to conclude $q=p$?

Write an equation of a line containing (2,4) and parallel to x−2y=5

Find the geometric means in the sequence 4,_,_,_,324

Find 3 consecutive odd integers where the sum is -141

Diophantine Inequality $4ax-<!--\; -\; -->y^2+z<0$
For fixed positive integer a and positive z in
$4ax-<!--\; -\; -->y^2+z<0,$
find how many integer solutions (x,y) there are as a function of $z\in <!--\; \in \; -->{\mathbb{Z}}_{+}$, subject to constraints $x\ge <!--\; \ge \; -->0$ and $a\ge <!--\; \ge \; -->y\ge <!--\; \ge \; -->0$.
This form comes from the quadratic polynomial discriminant. I am not sure where to begin as this isn't the usual quadratic Diophantine inequality (or equation for that matter) that I find through search engines.

Find the slope of a line parallel to the graph of each equation y−17=−3x

Find the common ratio for 2,−6,18,−54,162,...

Help with recognizing a geometric sequence

Find the sum of the arithmetic sequence having the data given ${\displaystyle a_1=7}$, d = - 3, n = 20

Are two lines with the same slope and y intercept considered to be parallel?

Write an equation of a line going through (3,-1) perpendicular to y=4x+1

Determine whether the sequence ${\displaystyle 1,\frac{1}{2},\frac{1}{3},\frac{1}{4},...}$ is geometric and if it is, what is the common ratio