Master Algebra 1 Word Problems with our Comprehensive Practice

Find the slope that is perpendicular to the line ${\displaystyle y=(-\frac{4}{5})x+1}$

I have two functions and want to find out in which points they intersect
$f(x)=-<!--\; -\; -->3x^3-<!--\; -\; -->3x^2+Ax$
$g(x)=x^3+x^2-<!--\; -\; -->6x$
I know that parameter should be $A=18$, but it was just my guess. What I want to as you for is the process how to find it.
I know that I must start by $f(x)=g(x)$ and then $f(x)-<!--\; -\; -->g(x)=0$ and so on, but I am stuck with quadratic equation with parameter.

Is the the derived set of irrational numbers $\subset <!--\; \subset \; -->\mathbb{R}$ an empty set?

The first three terms of a geometric series are 5,15,45. Find the sum of the first n terms if the nth term of this series is 10935

If $x,y,\mathrm{\alpha <!--\; \alpha \; -->}>0$ and $x>y$, then $x^{\mathrm{\alpha <!--\; \alpha \; -->}}>y^{\mathrm{\alpha <!--\; \alpha \; -->}}$
I know it's obvious when we use differnetation rule for exponential function, but I'm not allowed to. Is there any way I can show it clearly?

Let $a,b\in <!--\; \in \; -->\mathbb{R}$ with a<b and $f:[a,b]\to <!--\; \to \; -->[a,b]$ be a continuous function. If $a_0,a_1,...,a_n\in <!--\; \in \; -->\mathbb{R}$ and $a_0+a_1+a_2+...+a_n\ne <!--\; \ne \; -->0$ and $a_{0}x+a_{1}f(x)+a_2{f}^2(x)+...+a_n{f}^n(x)=0,\mathrm{\forall <!--\; \forall \; -->}x\in <!--\; \in \; -->[a,b]$, prove that $a\cdot <!--\; \cdot \; -->b<0$

Find the sum of a 52–term arithmetic sequence where the first term is 1 and the last term is 154

Find the formula for ${\displaystyle a_n}$ for the arithmetic sequence ${\displaystyle a_1=15,d=4}$

Exhibit a countable set of irrational numbers with justification

Write an equation of the line that passes through the point (2, 1) and has slope 5 in point-slope form

Find the slope of any line perpendicular to the line passing through (−9,8) and (−1,1)

Find the slope of any line perpendicular to the line passing through (2,17) and (8,−2)

Revenue Quadratic Functions
I am trying to determine a quadratic function to represent the following description.
A travel agenxy offers a group rate of $2400 per person for a week in London if 16 people sign up for the tour. For each additional person who signs up, the price per person is reduced $100. How many people, in total, must sign up for the tour in order for the travel agency to maximize their revenue? Determine the maximum revenue.
I know to solve the number of people would be to find the axis of symmetry, then for the maximum revenue it would be simply plugging that value into the equation to determine the max revenue.
But how do I set up the quadratic equation??

Sturm-Liouville problem - Partial Differential Equations
I was given the following Sturm-Liouville differential equation problem:
$x^2{y}^{″}+5x{y}^{\prime }+\mathrm{\lambda <!--\; \lambda \; -->}y=0$
$y(1)=y(e^{\mathrm{\pi <!--\; \pi \; -->}})=0$
I displayed the equation in the traditional Sturm-Liouville manner and found that the weight function equals $x^3$, but I am having trouble finding the eigenvalues for this problem.
We were told in class that because the coefficients of the ODE are functions of x and not constants, we should guess a solution for y in the form of $x^m$ and continue from there finding the eigenvalues.
Doing that I got a simple quadratic equation involving m:
$m^2-<!--\; -\; -->4m+\mathrm{\lambda <!--\; \lambda \; -->}=0$
But because it is a quadratic equation where p and q are both non zero I can't seem to find a way to divide the eigenvalues into the common categories which we usually did in class (for $\mathrm{\lambda <!--\; \lambda \; -->}<0,>0,=0$), like in this case for instance:
$m^2+\mathrm{\lambda <!--\; \lambda \; -->}=0$
I believe the solution is very simple but just can't find it, help would be much appreciated!

Find the indicated term of each arithmetic sequence ${\displaystyle a_1=46}$, d=5, n=14

Bella saw 7 out of 11 people brought a umbrella to the beach there were 99 people. how many of them did not have a beach umbrella?

Is 3, 6, 9, 12 ,15, 18 a geometric sequence?

Find an equation of a line containing the point (-2, 2), and parallel to the line 2(y + 1) = x

System of equations:
$$\left(\begin{array}{c}{\stackrel{\dot{}<!--\; \dot{}\; -->}{y}}_1\\ {\stackrel{\dot{}<!--\; \dot{}\; -->}{y}}_2\end{array}\right)=\left(\begin{array}{cc}2& 0\\ 4& -<!--\; -\; -->1\end{array}\right)\left(\begin{array}{c}{y}_1\\ y_2\end{array}\right)$$
Looking at matrix $A$ I can see a value not on the diagonal equal to zero. So the eigenvalues here are $2$ and $-<!--\; -\; -->1$, meaning I have a saddle node. Next I look at the eigenvectors:
$\mathrm{\lambda <!--\; \lambda \; -->}=2,\left[\begin{array}{ccc}0& 0& 0\\ 4& -<!--\; -\; -->3& 0\end{array}\right]$
$\mathrm{\lambda <!--\; \lambda \; -->}=-<!--\; -\; -->1,\left[\begin{array}{ccc}3& 0& 0\\ 4& 0& 0\end{array}\right]$
$y=A{e}^{2x}\left(\begin{array}{c}3\\ 4\end{array}\right)+B{e}^{-<!--\; -\; -->x}\left(\begin{array}{c}0\\ 1\end{array}\right)$
So I have an attractive trajectory on the $y$-axis, and a repulsive trajectory on the line $y=\frac{3}{4}x$
With the slope at origin being
$\frac{{\stackrel{\dot{}<!--\; \dot{}\; -->}{y}}_2}{{\stackrel{\dot{}<!--\; \dot{}\; -->}{y}}_1}=\frac{4y_1-<!--\; -\; -->y_2}{2y_1}=\frac{0}{0}=\pm <!--\; \pm \; -->\mathrm{\infty <!--\; \infty \; -->}$
Does all of this seem correct?

Find the 9th term of the geometric sequence –1, 3, –9, …