Master Algebra 1 Word Problems with our Comprehensive Practice

Find a standard form equation for the line with A(2, −4) and is parallel to the line 5x-2y=4

Find a1 given an = 24, r = 2/3, n = 4

A Mighty Kids meal includes 3.2 ounces of applesauce. If a child eats two frozen meals each week, how many ounces of applesauce will they consume in one year?

Find the integers if the sum of three consecutive odd integers is 177

I'm interested in solving a heat equation with a discontinuous source term in one dimension on the real line:
$\frac{\mathrm{\partial <!--\; \partial \; -->}u}{\mathrm{\partial <!--\; \partial \; -->}t}(x,t)=-<!--\; -\; -->x\mathrm{\theta <!--\; \theta \; -->}(x)u(x,t)-<!--\; -\; -->x\mathrm{\theta <!--\; \theta \; -->}(-<!--\; -\; -->x)u(-<!--\; -\; -->x,t)+D\frac{{\mathrm{\partial <!--\; \partial \; -->}}^{2}u}{\mathrm{\partial <!--\; \partial \; -->}{x}^2}(x,t)$
where $\mathrm{\theta <!--\; \theta \; -->}(\cdot <!--\; \cdot \; -->)$ is the Heaviside function. The boundary conditions are that $\underset{|x|\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}u(x,t)=0$ The initial condition is arbitrary, but even for a symmetric Gaussian or a symmetric exponential an analytical solution would be interesting.
Because of the discontinuity in the right-hand side induced by $\mathrm{\theta <!--\; \theta \; -->}(\cdot <!--\; \cdot \; -->)$, my intuition would be to solve in the two domains x>0 and x<0 and patch them together at x=0. However, the source terms have the effect of causing exponential decay with rate x at the point x for x>0 and corresponding growth at the point −x. This introduces an asymmetry, which should cause a diffusive flux back to the side x>0. Solving for the solution in the two domains does not seem to capture this feedback, and I'm not sure how to solve globally. What are some avenues to proceed?

$a:[0,\mathrm{\infty <!--\; \infty \; -->})\to <!--\; \to \; -->\mathbb{R}$ is a continous and bounded and
$x^{\prime }(t)=\left(\begin{array}{cc}0& 1\\ -<!--\; -\; -->a(t)& 0\end{array}\right)x(t)$
has a non-zero solution like $y(t)$ such that $\underset{t\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}y(t)=0$.
Show that this equation has an unbounded solution on $[0,\mathrm{\infty <!--\; \infty \; -->})$.

Is the following a geometric sequence 1.5, 2.25, 3.375, 5.0625,…

Find the slope of the line perpendicular to ${\displaystyle y=\frac{8}{5}x-3}$

For what class of functions is steepest descent guaranteed to converge in finite number of iterations?
Is there any classification of ${\mathbb{R}}^n\to <!--\; \to \; -->\mathbb{R}$ functions with respect to finite convergence of steepest descent (and similar first-order methods, e.g., nonlinear conjugate gradients)?
By finite convergence I mean that algorithm reaches a point $x_k$ with $||\mathrm{\nabla <!--\; \nabla \; -->}f(x_k)||=0$ after a finite number of iterations, given that no round-off errors occur (and maybe assuming that line search is exact).
It is known that algorithms are globally convergent, i.e. $li{m}_{k\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}||\mathrm{\nabla <!--\; \nabla \; -->}f(x_k)||=0$, but there is no guarantee that it happens for finite k.
For example, conjugate gradient method is guaranteed to converge in n steps for positive definite quadratic functions. Are there any similar results for steepest descent?
What about (strongly) convex general functions? I was only able to find results of the type
$k=O\left(\frac{||x_0-<!--\; -\; -->x^{\ast <!--\; \ast \; -->}|{|}^2}{\mathrm{ϵ<!--\; ϵ\; -->}}\right)$
to achieve $f(x_k)-<!--\; -\; -->f(x^{\ast <!--\; \ast \; -->})\le <!--\; \le \; -->\mathrm{ϵ<!--\; ϵ\; -->}$, where f is smooth convex and x∗ is a stationary point.
As far as I understand, it does not imply finite convergence for $\mathrm{ϵ<!--\; ϵ\; -->}=0$. Is it known that finite convergence does not happen even for strongly convex functions?

Are there many more irrational numbers than rational?

Find answers of this system of equations in real numbers
$\{\begin{array}{c}x+\frac{2}{y}=3\\ y+\frac{2}{z}=3\\ z+\frac{2}{x}=3\end{array}$

If $a,b,m$ and $n$ are positive integers such that $\sqrt[m]{a}$ and $\sqrt[n]{b}$ are irrational numbers, how can we prove that the sum $\sqrt[m]{a}+\sqrt[n]{b}$ is also irrational?

Suppose $\gamma$ is a kth root of unity that satisfies a quadratic equation $z^2-mz-n=0$ with $m,n\in <!--\; \in \; -->\mathbb{Z}$. Then $k=3,4$ or 6.
Let $k\in <!--\; \in \; -->\mathbb{Z}$ with $k>2$ and suppose $\mathrm{\gamma <!--\; \gamma \; -->}$ is a kth root of unity that satisfies a quadratic equation $z^2-<!--\; -\; -->mz-<!--\; -\; -->n=0$ with $m,n\in <!--\; \in \; -->\mathbb{Z}$. Then $k=3,4$ or 6.
My knowledge on algebra and number theory is poor, I just need this lemma to complete the computation of the analytic automorphism groups of complex tori.Can you help me prove this lemma with the knowledge of field theory and Euler phi function as small as possible? Actually I forget almost everything on field theory and have never learned number theory.

For what value of k are the graphs of 12y=−3x+8 and 6y=kr−5 parallel? For what value of k are they perpendicular?

Correct notation for inequality equation. I'm currently taking a Polynomial functions -course and learning about quadratic inequality equations.
Let's consider this equation:
$-<!--\; -\; -->x^2+7x<10$
The correct answer to this according to our book is simply
$x<2$ or $x>5$
But is there any other way to express this in a more mathematical way? I understand why this is expressed the way it is as we haven't learned about set theory yet, but I'd like to know how it is expressed "correctly".

Find the next three terms of the arithmetic sequence 3,7,11,15,...

Finding the Maximum and Minimum of Trigonometric Equations By Quadratic Substitution
$y=3\cos <!--\; \; -->2x+7\sin <!--\; \; -->x+2$
The original question was: Finding the stationary point of the function. I tried to use both differentiation and trigonometric substitution of cos(2x) into $1-<!--\; -\; -->2{\sin }^2<!--\; \; -->(x)$.
For the quadratic method, I substituted sin(x) as u and solved for u, where the the equation is equated to 0, and finding the midpoint of the roots to find out the value of x following the nature of a parabolic equation.
Unfortunately I realized that trigonometric substitution eliminates a possible solution for x where $\cos <!--\; \; -->(x)=0$. I suspect that it might be due to a non function being converted into a function but I do not quite understand why does it causes so.

Is "$120\mathrm{\%<!--\; \%\; -->}$" an integer or irrational number ?

Find the sum of the geometric sequence -1, 6, -36, ... if there are 7 terms

How do you find ${\displaystyle a_{25}}$ given ${\displaystyle a_n={(-1)}^n(3n-2)}$?