Master Algebra 1 Word Problems with our Comprehensive Practice

Prove that every open interval in real numbers contains rational, irrational and dyadic numbers.

In other words, for an irrational number
$$x=\underset{n\in <!--\; \in \; -->\mathbb{Z}}{\sum <!--\; \sum \; -->}{a}_n{10}^n$$
Does $a_n$ takes all the numbers between $0$ and $9$ ?

Find the slope of any line perpendicular to the line passing through (−12,21) and (−18,1)

Find the slope of any line perpendicular to the line passing through (15,−12) and (24,27)

How would you show that if we have a sequence $\{a_n\}$ of irrational numbers then for any number $a$ in the reals, the $|a_n-<!--\; -\; -->a|<\mathrm{ϵ<!--\; ϵ\; -->}$ for $\mathrm{ϵ<!--\; ϵ\; -->}>0$.

I want to know are the graph y=−3x+7 and y=−3x+2 perpendicular

A pharmacist has a 10% alcohol solution and a 25% alcohol solution. So i want to know how many milliliters of each solution will she need to mix together in order to have 200 mL of a 20% alcohol solution?

Write an equation in point slope form when slope is ${\displaystyle \frac{1}{3}}$ and the y- intercept is –4

Write an equation in point-slope form for the given (1,2), m= 2

Find the slope of any line perpendicular to the line passing through (−2,32) and (1,5)

${\displaystyle 2x^3-5x^2<2x-5}$

Find the slope of the line perpendicular to ${\displaystyle y=\frac{5}{2}x-9}$

Finding the stationary point of a type of hyperbola?
I know that to find stationary points on a function, we need to differentiate the function and set that = 0.
But how can we find the stationary points of the below function?
$$y=\frac{1}{x}+\frac{1}{x^2}-<!--\; -\; -->\frac{1}{x^3}$$
When I differentiate it, I get fractions with x in the denominator... I try to factorize and I get a quadratic equation with non-real roots and then $\frac{1}{x^4}$ which $=0$. But the only way for $\frac{1}{x}=0$ is if $x=\mathrm{\infty <!--\; \infty \; -->}$.

How can I solve this system of equation?
1.9. The program should take three numbers: a; b; c and find the roots of the quadratic equation in the form:
$a{x}^2+bx+c=0$
If the value of the determinant of the quadratic equation is negative (i.e. $\mathrm{△<!--\; △\; -->}<0$), the program should write an appropriate message.
1.10. Modify the 1.9 so that the parameters A and a are functions of the parameter $\mathrm{\varphi <!--\; \varphi \; -->}$, e.g.
$A(\mathrm{\varphi <!--\; \varphi \; -->}=A(1.0+0.25\sin <!--\; \; -->(\mathrm{\varphi <!--\; \varphi \; -->}))$
$a(\mathrm{\varphi <!--\; \varphi \; -->})=a(1.0+0.5\cdot <!--\; \cdot \; -->|\sin <!--\; \; -->(\mathrm{\varphi <!--\; \varphi \; -->})|)$
In the program, declare functions for the above equations and save the results in an array.
I have solved the 1.9 as that was easy. But I am stuck with the following:
$\begin{array}{}\text{(1)}& f(x)=f(1+0.25\sin <!--\; \; -->(x))\end{array}$
$\begin{array}{}\text{(2)}& g(x)=g(1+0.5|\sin <!--\; \; -->(x)|)\end{array}$
My attempt:
Given,
$f(x)=f(1+0.25\sin <!--\; \; -->(x))$
Now,
$f(1+0.25\sin <!--\; \; -->(x))=\frac{a}{16}{\sin }^2<!--\; \; -->x+(\frac{a}{2}+\frac{b}{4})\sin <!--\; \; -->x+(a+b+c)$
Let, $\sin <!--\; \; -->x=y$
So, the above equation becomes: $y=\frac{-<!--\; -\; -->(2a+b)\pm <!--\; \pm \; -->\sqrt{b^2-<!--\; -\; -->4ac}}{2a}$
Hence,
$\begin{array}{rl}\sin <!--\; \; -->x& ={\displaystyle \frac{-<!--\; -\; -->(2a+b)\pm <!--\; \pm \; -->\sqrt{b^2-<!--\; -\; -->4ac}}{2a}}\\ x& ={\sin }^{-<!--\; -\; -->1}<!--\; \; -->\left({\displaystyle \frac{-<!--\; -\; -->(2a+b)\pm <!--\; \pm \; -->\sqrt{b^2-<!--\; -\; -->4ac}}{2a}}\right)\end{array}$
Are this procedure and solution correct?

When does this least squares analytical solution based on zeros of partial derivatives start providing more than one solution?
If I want to fit a quadratic function of two variables to some data, I can use
$$f(x,y)=c_1{x}^2+c_{2}xy+c_3{y}^2+c_{4}x+c_{5}y+c_6$$
$$\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}{c}_i}\underset{j}{\sum <!--\; \sum \; -->}{(z_j-<!--\; -\; -->f(x_j,y_j))}^2=0$$
to obtain six equations, and then endeavor to solve them.
I've done it for one variable not two, but I'm guessing the process is straightforward.
If I extend this to more variables, and to higher order than 2, when will the analytical expressions be at risk for having multiple solutions?

Critical Points of Quadratic Forms
Just a question about finding critical points (points where the differential is not surjective). I have the equation
$$f(x)=x^{t}Ax$$
where A is a symmetric n by n matrix and x is an element of ${\mathbb{R}}^n$. Now, I've found the derivative of the function to be
$$df(x)=2x^{t}A=2Ax$$
by symmetry of A. I'm having trouble determining values in ${\mathbb{R}}^n$ where the derivative is not surjective. It's clear that for $x=(0,...,0)$, the derivative is not surjective, but my intuition is that for any other value of x, and for any value in ${\mathbb{R}}^n$, we can choose a corresponding A. Am I thinking about this correctly? Is my derivative correct for the quadratic form $f(x)=x^{t}Ax$?

I want to find the slope that is perpendicular to the line 3x+10y=−20

Find nth term rule for 1,3,9,27,...

Find the sum of the geometric sequence -4,24,-144,... if there are 8 terms

Find the tenth term of the geometric sequence with a4 = 16 and r = 2