Master Algebra 1 Word Problems with our Comprehensive Practice

For which numbers a,b,c and d will the function $f(x)=\frac{(ax+b)}{(cx+d)}$ satisfy $f(f(x))=x$ for all x?

Find the slope of a line parallel to the line ${\displaystyle y=\frac{1}{3}x+2}$

Two angles of a triangle measure 15 and 85. Find the measure for the third angle

The length of a picture frame is 3 in. greater than the width. The perimeter is less than 52 in. Find the dimensions of the frame

How do you write the first five terms of the sequence ${\displaystyle a_n=5n-3}$?

$S=\{x\in <!--\; \in \; -->\mathbb{R}\mathrm{\setminus <!--\; \setminus \; -->}\mathbb{Q}:0\le <!--\; \le \; -->x\le <!--\; \le \; -->1\}$ . How would you prove if $S$ is closed or not. I know that there are infinitely many irrational numbers between $0$ and $1$, and that $Q$ is dense in $R$, but would that help?

Find the larger of the two numbers if the sum of two numbers is 41, one number is less than twice the other.

The basketball team scored 12 points the first game and 40 points the eighth game of the season. How many points will they earn with this pattern on the 13th game of the season?

Find the next three terms of the arithmetic sequence −2,−5,−8,−11,...

Quadratic Variation in SDEs.
Let $X_t$ be a stochastic process satisfying:
${\displaystyle X_t=X_0+{\int <!--\; \int \; -->}_0^t\mathrm{\mu <!--\; \mu \; -->}(s,\mathrm{\omega <!--\; \omega \; -->})\mathrm{d}s+{\int <!--\; \int \; -->}_0^t\mathrm{\nu <!--\; \nu \; -->}(s,\mathrm{\omega <!--\; \omega \; -->})\mathrm{d}{B}_s}$
Shorthand: $\mathrm{d}{X}_t={\mathrm{\mu <!--\; \mu \; -->}}_t\mathrm{d}t+{\mathrm{\nu <!--\; \nu \; -->}}_t\mathrm{d}{B}_t$
where the last integral is a Brownian motion integral, and where $\mathrm{\mu <!--\; \mu \; -->}(t,\mathrm{\omega <!--\; \omega \; -->}),\mathrm{\nu <!--\; \nu \; -->}(t,\mathrm{\omega <!--\; \omega \; -->})$ are ${\mathcal{F}}_t$ adapted $L^2$ functions. (In class we so far only did the case where $\mathrm{\mu <!--\; \mu \; -->}$ and $\mathrm{\nu <!--\; \nu \; -->}$ are deterministic).
Let f(t,x) be a twice differentiable deterministic function.
The usual presentation of Ito's lemma is:
${\displaystyle \mathrm{d}f(t,X_t)=\left(\frac{\mathrm{\partial <!--\; \partial \; -->}f}{\mathrm{\partial <!--\; \partial \; -->}t}+{\mathrm{\mu <!--\; \mu \; -->}}_t\frac{\mathrm{\partial <!--\; \partial \; -->}f}{\mathrm{\partial <!--\; \partial \; -->}x}+\frac{1}{2}\frac{{\mathrm{\partial <!--\; \partial \; -->}}^{2}f}{\mathrm{\partial <!--\; \partial \; -->}{x}^2}{\mathrm{\nu <!--\; \nu \; -->}}_t^2\right)\mathrm{d}t+\frac{\mathrm{\partial <!--\; \partial \; -->}f}{\mathrm{\partial <!--\; \partial \; -->}x}{\mathrm{\nu <!--\; \nu \; -->}}_t\mathrm{d}{B}_t}$
The professor offered us this shorthand:
${\displaystyle \mathrm{d}f(t,X_t)=\frac{\mathrm{\partial <!--\; \partial \; -->}f}{\mathrm{\partial <!--\; \partial \; -->}t}\mathrm{d}t+\frac{\mathrm{\partial <!--\; \partial \; -->}f}{\mathrm{\partial <!--\; \partial \; -->}x}\mathrm{d}{X}_t+\frac{1}{2}\frac{{\mathrm{\partial <!--\; \partial \; -->}}^{2}f}{{\mathrm{\partial <!--\; \partial \; -->}}^{2}x}(\mathrm{d}{X}_t{)}^2}$
He explained that the notation $(\mathrm{d}{X}_t{)}^2$ is to be interpreted as the taking the quadratic variation whenever the algebra would suggest you multiply the differentials. For example, $\mathrm{d}{B}_t\mathrm{d}{B}_t=\mathrm{d}⟨<!--\; ⟨\; -->B_t,B_t{⟩<!--\; ⟩\; -->}_T=\mathrm{d}(T)\text{a.s.}=\mathrm{d}t$ (this last step has its own answer on s.e.) Why is the formal multiplication of (stochastic) differentials interpreted as the quadratic variation?

A line perpendicular to y = 1/4x + 3 passes through the point (0,6). Which other point lies on the line?

Write the equation of a line in slope intercept form that is perpendicular to the line y = –4x and passes through the point (2, 6)

Consider the modification to the Malthusian equation
$$\frac{dN}{dt}=rS(N)N,$$
where r>0 is the per capita growth rate, and S(N) is a survival fraction. For some organisms, finding a mate at low population densities may be difficult. In such cases, the survival fraction can take the form $S(N)=\frac{N}{A+N}$, where A>0 is a constant.
(a)
i. By examining what happens to S(N) for N≫A and N≪A explain why this fraction models the situation outlined above.
Attempt:
When N≫A (my assumption is that the sign '≫'means significantly greater than ) so A is much smaller than N then the constant A.This would effect the survival fraction in a way which it becomes roughly equal to 1 since A is really small.
When N≪A (my assumption is that the sign '≪' means significantly smaller than) so N is much smaller than A will impact the survival fraction in a way which will make it become small.
The survival fraction models matches with the situation above since:
For a low population finding a mate is difficult (a.k.a N≪A )
For a large population finding a mate is more likely (a.k.a N≫A )
ii. By examining the form of the equation, determine the long-term behaviour of a population for an initial condition N(0)>0.
Attempt: For the initial conditions stated above the Malthusian equation has the form:
$$\frac{dN}{dt}=r\frac{N}{A+N}N,$$
which has the solution.
$$N(t)=N_0{e}^{\mathrm{\lambda <!--\; \lambda \; -->}t}$$
If $\mathrm{\lambda <!--\; \lambda \; -->}>0$ then exponential growth
If $\mathrm{\lambda <!--\; \lambda \; -->}<0$ then exponential decay
Please could you check if this is correct. If it is please can you suggest any improvement I could add to my attempts.

If two periodic functions so that $X=X(t)$ and $Y=X(t+\mathrm{\alpha <!--\; \alpha \; -->})$, with α unknown is there a way to extract $$\mathrm{\alpha <!--\; \alpha \; -->}$$ from $X-<!--\; -\; -->Y$?

What are all tuples off postive integers $$w,x,y,z$$ that fullfill following system of equations:
$$x+10z^2=2014$$
$$2y+z=54$$
$$(y+2x+\frac{7}{2}w)z=1211$$

Find three consecutive odd integers that have a sum of 123

If the first term in a geometric sequence is 7, and the third term is 63, what is the ninth term?

Given that $$f(x)=4x-<!--\; -\; -->x^2$$ for $$x\le <!--\; \le \; -->2$$ is invertible, find $$f^{-<!--\; -\; -->1}(x)$$

Find three consecutive integers whose sum is 42

Calculate the perimeter of a square if the area is 81